2007-07-24 17:45:46 +00:00
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#LyX 1.5.0svn created this file. For more info see http://www.lyx.org/
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\lyxformat 276
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2006-11-01 08:41:31 +00:00
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2006-10-31 20:56:42 +00:00
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2006-11-01 10:06:31 +00:00
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2006-10-31 20:56:42 +00:00
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%\pgffill
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}
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}
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\end_preamble
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\options notes=show
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\language english
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\inputencoding auto
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\paragraph_separation indent
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\quotes_language english
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\paperpagestyle default
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\output_changes false
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\end_header
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\begin_body
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\begin_layout Title
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The Complexity of
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\newline
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Finding Paths in Tournaments
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\end_layout
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\begin_layout Author
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Till Tantau
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\end_layout
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\begin_layout Institute
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International Computer Schience Institute
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\newline
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Berkeley, California
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\begin_inset OptArg
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status collapsed
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\begin_layout Standard
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ICSI
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Date
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January 30th, 2004
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\end_layout
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\begin_layout BeginFrame
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Outline
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\end_layout
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\begin_layout Standard
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\begin_inset LatexCommand tableofcontents
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\end_inset
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\begin_inset ERT
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status collapsed
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\begin_layout Standard
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[pausesections]
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\end_layout
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\end_inset
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\end_layout
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\begin_layout EndFrame
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\end_layout
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\begin_layout Standard
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status collapsed
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\begin_layout Standard
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% Show the table of contents at the beginning
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\end_layout
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\begin_layout Standard
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% of every subsection.
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\end_layout
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\begin_layout Standard
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\begin_layout Standard
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\backslash
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AtBeginSubsection[]{
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\end_layout
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\backslash
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frame<handout:0>{
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\end_layout
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\begin_layout Standard
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\backslash
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frametitle{Outline}
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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tableofcontents[current,currentsubsection]
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Section
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2006-10-31 20:56:42 +00:00
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Introduction
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Subsection
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2006-10-31 20:56:42 +00:00
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What are Tournaments?
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout BeginFrame
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2006-10-31 20:56:42 +00:00
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Tournaments Consist of Jousts Between Knights
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Columns
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\end_layout
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2006-11-01 08:41:31 +00:00
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\begin_deeper
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\begin_layout Column
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5.75cm
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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\begin_inset ERT
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2006-11-01 08:41:31 +00:00
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status inlined
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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begin{pgfpicture}{1.25cm}{-1cm}{7cm}{4cm}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodebox{A}[virtual]{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(2,1)}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfuseimage{knight1}}{2pt}{2pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodebox{B}[virtual]{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(6,1)}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfuseimage{knight2}}{2pt}{2pt}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodebox{C}[virtual]{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(4,-1)}{
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfuseimage{knight3}}{2pt}{2pt}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodebox{D}[virtual]{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(4,3)}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfuseimage{knight4}}{2pt}{2pt}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\begin_layout Standard
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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color{beamerexample}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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only<3->{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetendarrow{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfarrowto}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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only<2->{
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfsetlinewidth{0.6pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A}{B}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A}{C}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{D}{A}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C}{B}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B}{D}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C}{D}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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end{pgfpicture}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Column
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6cm
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Block
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
{What is a Tournament?}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_deeper
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\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status collapsed
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
<1->
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
|
|
A group of knights.
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
<2->
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
Every pair has a joust.
|
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
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|
<3->
|
2006-11-01 08:41:31 +00:00
|
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\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
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In every joust one knight wins.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
|
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\end_deeper
|
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\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
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Tournaments are Complete Directed Graphs
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Columns
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_deeper
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\begin_layout Column
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5cm
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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color{beamerexample}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetlinewidth{0.6pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{A}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(2.5,1)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{B}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(5.5,1)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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graphnode{C}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(4,-0.5)}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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graphnode{D}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfxy(4,2.5)}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
color{white}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfnodecenter{A}}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[center,center]{$v_2$}}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodecenter{B}}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[center,center]{$v_3$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodecenter{C}}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[center,center]{$v_4$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodecenter{D}}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[center,center]{$v_1$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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\begin_layout Standard
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{beamerexample}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetendarrow{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfarrowto}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfnodesetsepstart{2pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfnodesetsepend{4pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A}{B}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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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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|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfnodeconnline{A}{C}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
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|
\end_layout
|
|
|
|
|
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|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D}{A}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
|
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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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|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C}{B}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
|
|
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|
\begin_layout Standard
|
|
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|
|
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|
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|
\end_layout
|
|
|
|
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|
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|
\begin_layout Standard
|
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|
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|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfnodeconnline{B}{D}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D}{C}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
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|
\begin_layout Standard
|
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|
|
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|
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|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
end{pgfpicture}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Column
|
|
|
|
|
6cm
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Definition
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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<2->
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2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
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2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
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A
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\color none
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2006-10-31 20:56:42 +00:00
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\color red
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tournament
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2007-07-24 17:45:46 +00:00
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\color none
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2006-11-01 09:22:35 +00:00
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\color inherit
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2007-07-24 17:45:46 +00:00
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is a
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_deeper
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\begin_layout Enumerate
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2006-10-31 20:56:42 +00:00
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directed graphs,
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Enumerate
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2006-10-31 20:56:42 +00:00
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with exactly one edge between
|
2006-11-01 08:41:31 +00:00
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\newline
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2006-10-31 20:56:42 +00:00
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any two different vertices.
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
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\end_deeper
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\begin_layout BeginFrame
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2006-10-31 20:56:42 +00:00
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\begin_inset ERT
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2006-11-01 08:41:31 +00:00
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status collapsed
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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[<+>]
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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Tournaments Arise Naturally in Different Situations
|
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout ExampleBlock
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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{Applicatins in Ordering Theory}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_deeper
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
Elements in a set need to be sorted.
|
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|
2006-11-01 08:41:31 +00:00
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\newline
|
2006-10-31 20:56:42 +00:00
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|
The comparison relation may be cyclic, however.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
|
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\begin_layout Separator
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout ExampleBlock
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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status inlined
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
{Applications in Sociology}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_deeper
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
Several candidates apply for a position.
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
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|
Reviewers decide for any two candidates
|
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|
whom they prefer.
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_deeper
|
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|
\begin_layout Separator
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\end_layout
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout ExampleBlock
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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|
status inlined
|
|
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|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
|
{Applications in Structural Complexity Theory}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
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|
\begin_deeper
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
A language
|
|
|
|
|
\begin_inset Formula $L$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
is given and a selector function
|
|
|
|
|
\begin_inset Formula $f$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
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|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
It chooses from any two words the one more likely to be in
|
|
|
|
|
\begin_inset Formula $f$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
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|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Subsection
|
2006-10-31 20:56:42 +00:00
|
|
|
What Does ``Finding Paths'' Mean?
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
``Finding Paths'' is Ambiguous
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
|
{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
strut Input for
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
def
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
par{}% because LyX inserts superfluous paragraphs
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
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|
\begin_layout Standard
|
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|
|
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|
\end_layout
|
|
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|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<1>{Path Finding Problems}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
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|
|
|
\end_layout
|
|
|
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|
|
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|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
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|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<2-3>{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}$}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
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|
|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
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|
\end_layout
|
|
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|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<4-5>{the Construction Problem}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
|
|
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|
\begin_layout Standard
|
|
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|
|
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|
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|
\end_layout
|
|
|
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|
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<6-7>{the Optimization Problem}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
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|
\begin_layout Standard
|
|
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|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<8-9>{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{distance}$}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<10->{the Approximation Problem}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
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|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2007-07-24 17:45:46 +00:00
|
|
|
A
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
graph
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
2006-10-31 20:56:42 +00:00
|
|
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|
2007-07-24 17:45:46 +00:00
|
|
|
\color inherit
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $G=(V,E)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
, a
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
source
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
\color inherit
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $s\in V$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
and a
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
target
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
\color inherit
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $t\in V$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<only@-9| visible@8->
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
A
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
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|
maximum distance
|
2006-11-01 09:22:35 +00:00
|
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|
\color inherit
|
2006-11-01 08:41:31 +00:00
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|
\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset Formula $d$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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.
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
phantom{p}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
<only@10->
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
|
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An
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\color none
|
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|
2006-10-31 20:56:42 +00:00
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\color red
|
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|
approximation ratio
|
2007-07-24 17:45:46 +00:00
|
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\color none
|
2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
|
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\color inherit
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|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset Formula $r>1$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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\end_deeper
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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|
status collapsed
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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nointerlineskip
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Overprint
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_deeper
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
onslide<1,3,5,7,9,11-12>
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Columns
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
[t,onlytextwidth]
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
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|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
alt<1-2>{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
column{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
textwidth}}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
column{5cm}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout ExampleBlock
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
|
|
|
{Example Input}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
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|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{beamerexample}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetlinewidth{0.6pt}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
graphnode{A}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxy(3,1)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
graphnode{B}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxy(5,1)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
graphnode{C}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxy(4,0)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
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|
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|
\begin_layout Standard
|
|
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|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
graphnode{D}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxy(4,2)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
|
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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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|
\begin_layout Standard
|
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{white}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodecenter{B}}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[center,center]{$t$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
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|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
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|
|
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|
\begin_layout Standard
|
|
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|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodecenter{D}}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[center,center]{$s$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
|
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|
|
|
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|
|
|
\end_layout
|
|
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|
|
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|
\begin_layout Standard
|
|
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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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|
|
|
|
|
|
\end_layout
|
|
|
|
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|
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{beamerexample}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
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|
\begin_layout Standard
|
|
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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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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetendarrow{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfarrowto}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
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|
|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
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|
\end_layout
|
|
|
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|
\begin_layout Standard
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodesetsepstart{2pt}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodesetsepend{4pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A}{B}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A}{C}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{D}{A}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C}{B}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B}{D}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{D}{C}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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only<9> {
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfxy(5.3,1)}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfbox[left,center]{, $d=2$}}}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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only<11>{
|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfxy(5.3,1)}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[left,center]{, $r=1.5$}}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
|
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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only<12>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfxy(5.3,1)}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[left,center]{, $r=1.25$}}}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
end{pgfpicture}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\end_inset
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_deeper
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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status inlined
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
only<3->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
column{5cm}}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout ExampleBlock
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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status inlined
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
<only@3->{Example Output}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_deeper
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<5-8,10->{
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{beamerexample}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetlinewidth{0.6pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
graphnode{A}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfxy(3,1)}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
graphnode{B}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfxy(5,1)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
graphnode{C}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxy(4,0)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
graphnode{D}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxy(4,2)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
|
|
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|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{white}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
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|
|
|
\begin_layout Standard
|
|
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|
|
|
|
|
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|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
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|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodecenter{B}}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[center,center]{$t$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
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|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfputat{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
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pgfnodecenter{D}}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfbox[center,center]{$s$}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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2006-10-31 20:56:42 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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color{beamerexample}
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfsetendarrow{
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\backslash
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pgfarrowto}
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodesetsepstart{2pt}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodesetsepend{4pt}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<7,12>{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A}{B}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<5,11>{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A}{C}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<5,7,11-12>{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{D}{A}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<5,11>{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C}{B}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B}{D}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{D}{C}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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only<3,9>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(2.75,1)}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfbox[left,center]{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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alert{``Yes''}}}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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\backslash
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2006-10-31 20:56:42 +00:00
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end{pgfpicture}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
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\end_deeper
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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onslide<2,4,6,8,10>
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Block
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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{Variants of Path Finding Problems}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_deeper
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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usedescriptionitemofwidthas{Approximation Problem:}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Description
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Reachability\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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Problem:
|
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status collapsed
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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<2->
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
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|
Is there a path from
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\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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to\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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?
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Description
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Construction\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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Problem:
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status collapsed
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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<4->
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
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Construct a path from
|
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\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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to\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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?
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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|
\begin_layout Description
|
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Optimization\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
Problem:
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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|
status collapsed
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
<6->
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
Construct a shortest path from
|
|
|
|
|
\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
|
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\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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|
to\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
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\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
|
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\end_inset
|
2006-10-31 20:56:42 +00:00
|
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Description
|
|
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|
Distance\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
Problem:
|
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
<8->
|
2006-11-01 08:41:31 +00:00
|
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\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
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Is the distance of
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\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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and\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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at most\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $d$
|
2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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?
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Description
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Approximation\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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Problem:
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status collapsed
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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<10->
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2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
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Construct a path from
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\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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to\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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of length
|
2006-11-01 08:41:31 +00:00
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\newline
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2006-10-31 20:56:42 +00:00
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approximately their distance.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
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\end_deeper
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\begin_layout Section
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2006-10-31 20:56:42 +00:00
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Review
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Subsection
|
2006-10-31 20:56:42 +00:00
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Standard Complexity Classes
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfdeclaremask{computer-mask}{beamer-g4-mask}
|
2006-11-01 08:41:31 +00:00
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\backslash
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pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer
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-g4}
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
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The Classes L and NL are Defined via
|
2006-11-01 08:41:31 +00:00
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\newline
|
2006-10-31 20:56:42 +00:00
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Logspace Turing Machines
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status open
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(0,4)}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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uncover<2->{
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(0,0.5)}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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tape{}{output tape (write only)}{10690836937182}}}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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uncover<3->{
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(7,2)}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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shorttape{work tape (read/write), $O(
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
log n)$ symbols}{}{42}}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\backslash
|
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pgfputat{
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\backslash
|
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pgfxy(1.75,2.5)}{
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfbox[center,center]{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfuseimage{computer}}}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
|
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|
}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetlinewidth{0.6pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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color{structure}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetendarrow{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfarrowto}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
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|
pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85)
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
uncover<2->{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
uncover<3->{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Standard
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
end{pgfpicture}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
|
|
|
|
Logspace Turing Machines Are Quite Powerful
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{Deterministic logspace machines can compute}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
addition, multiplication, and even division
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
reductions used in completeness proofs,
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
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|
reachability in forests.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
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\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Block
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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{Non-deterministic logspace machines can compute}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_deeper
|
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\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
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reachability in graphs,
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Itemize
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2006-10-31 20:56:42 +00:00
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non-reachability in graphs,
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
|
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2006-11-01 08:41:31 +00:00
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\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
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satisfiability with two literals per clause.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
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\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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<1>[label=hierarchy]
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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|
The Complexity Class Hierarchy
|
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\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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status inlined
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetlinewidth{0.8pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
heap{5.5}{3.5}{$
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
Class P$}{black}{1}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetdash{{2pt}}{0pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<2->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
heap{4.5}{3}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Class{NC}^2$}{black!50!structure}{2}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
heap{3.5}{2.5}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Class{NL}$}{black!50!structure}{3}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
heap{2.5}{2}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Class{L}$}{black!50!structure}{4}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<2->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
heap{1.75}{1.5}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
vphantom{A}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
smash{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Class{NC}^1}$}{black!50!structure}{5}}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
|
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetdash{}{0pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<2->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
heap{1.1}{1}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
vphantom{A}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
smash{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Class{AC}^0}$}{black}{6}}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetlinewidth{1.0pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{black}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxyline(-5,0)(5,0)
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<1-2>{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langat{3.375}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<1-2>{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langat{2.375}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{forest}}$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
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|
|
|
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|
\begin_layout Standard
|
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|
|
|
|
|
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|
\end_layout
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
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|
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|
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|
\end_layout
|
|
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|
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|
\begin_layout Standard
|
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|
|
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|
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|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<2>{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langat{0.975}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{addition}$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
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|
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|
\begin_layout Standard
|
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|
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|
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|
\end_layout
|
|
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|
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|
|
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|
\begin_layout Standard
|
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|
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|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<2>{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langatother{1.6}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
vbox{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{division}$,}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{parity}$}}}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<3-5>{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langat{3.375}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
vbox{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{distance}$,}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}$}}}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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|
\begin_layout Standard
|
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|
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|
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|
\end_layout
|
|
|
|
|
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|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<4->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langatother{2.375}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
vbox{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
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|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{distance}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{forest}}$,}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
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|
|
\begin_layout Standard
|
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|
|
|
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|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{forest}}$,}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{distance}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{path}}$,}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{path}}$}}}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<5->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langat{0.975}{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{tourn}}$}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<6->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langat{3.375}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
vbox{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{distance}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{tourn}}$,}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{distance}$,}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
ignorespaces
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hbox{$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{reach}$}}}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<7->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetdash{{1pt}}{0pt}
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
langat{2.375}{``$
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
Lang{approx}_{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
operatorname{tourn}}$''}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
end{pgfpicture}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
The Circuit Complexity Classes AC
|
|
|
|
|
\begin_inset Formula $^{0}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
, NC
|
|
|
|
|
\begin_inset Formula $^{1}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
, and NC
|
|
|
|
|
\begin_inset Formula $^{2}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
Limit the Circuit Depth
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
setlength
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
leftmargini{1em}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
nointerlineskip
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Columns
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
[t]
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Column
|
|
|
|
|
3.6cm
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Circuit Class
|
|
|
|
|
\begin_inset Formula $\Class{AC}^{0}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $O(1)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
depth
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
unbounded fan-in
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
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|
\begin_layout Examples
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{addition}\in\Class{AC}^{0}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{parity}\notin\Class{AC}^{0}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Column
|
|
|
|
|
3.6cm
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Circuit Class
|
|
|
|
|
\begin_inset Formula $\Class{NC}^{1}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $O(\log n)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
depth
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
bounded fan-in
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Examples
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{parity}\in\Class{NC}^{1}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{mutiply}\in\Class{NC}^{1}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{divide}\in\Class{NC}^{1}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Column
|
|
|
|
|
3.6cm
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Circuit Class
|
|
|
|
|
\begin_inset Formula $\Class{NC}^{2}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $O(\log^{2}n)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
depth
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
bounded fan-in
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Examples
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Class{NL}\subseteq\Class{NC}^{2}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout AgainFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<2>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
hierarchy
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Subsection
|
2006-10-31 20:56:42 +00:00
|
|
|
Standard Complexity Results on Finding Paths
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
All Variants of Finding Paths in Directed Graphs
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
Are Equally Difficult
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Fact
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{reach}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
and
|
|
|
|
|
\begin_inset Formula $\Lang{distance}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
are
|
|
|
|
|
\begin_inset Formula $\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
-complete.
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Corollary
|
2006-10-31 20:56:42 +00:00
|
|
|
For directed graphs, we can solve
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
the reachability problem in logspace iff
|
|
|
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
the construction problem in logspace iff
|
|
|
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
the optimization problem in logspace iff
|
|
|
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
the approximation problem in logspace iff
|
|
|
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout AgainFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<3>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
hierarchy
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
FindingPaths in Forests and Directed Paths is Easy,
|
2006-11-01 08:41:31 +00:00
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\newline
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2006-10-31 20:56:42 +00:00
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But Not Trivial
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Fact
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $\Lang{reach}_{\operatorname{forest}}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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and
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\begin_inset Formula $\Lang{distance}_{\operatorname{forest}}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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are
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\begin_inset Formula $\Class{L}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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-complete.
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Separator
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Fact
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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and
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\begin_inset Formula $\Lang{distance}_{\operatorname{path}}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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are
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\begin_inset Formula $\Class{L}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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-complete.
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout AgainFrame
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2006-10-31 20:56:42 +00:00
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\begin_inset ERT
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2006-11-01 08:41:31 +00:00
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status collapsed
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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<4>
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2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
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2006-10-31 20:56:42 +00:00
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hierarchy
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Section
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2006-10-31 20:56:42 +00:00
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Finding Paths in Tournaments
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Subsection
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2006-10-31 20:56:42 +00:00
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Complexity of: Does a Path Exist?
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout BeginFrame
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2006-10-31 20:56:42 +00:00
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Definition of the Tournament Reachability Problem
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Definition
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2007-07-24 17:45:46 +00:00
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Let
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\color none
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2006-10-31 20:56:42 +00:00
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\color red
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\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
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\color none
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2006-11-01 09:22:35 +00:00
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\color inherit
|
2007-07-24 17:45:46 +00:00
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contain all triples
|
2006-10-31 20:56:42 +00:00
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\begin_inset Formula $(T,s,t)$
|
2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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such that
|
2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_deeper
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\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
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\begin_inset Formula $T=(V,E)$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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is a tournament and
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Enumerate
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there exists a path from\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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to\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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|
\end_deeper
|
|
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\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
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|
The Tournament Reachability Problem is Very Easy
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Theorem
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout AlertBlock
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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|
status inlined
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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{Implications}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
|
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\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
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|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
The problem is
|
|
|
|
|
\begin_inset Quotes eld
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
easier
|
|
|
|
|
\begin_inset Quotes erd
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
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|
than
|
|
|
|
|
\begin_inset Formula $\Lang{reach}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
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|
|
and even
|
|
|
|
|
\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
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|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
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|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout AgainFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<5>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
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|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
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|
hierarchy
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Subsection
|
2006-10-31 20:56:42 +00:00
|
|
|
Complexity of: Construct a Shortest Path
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
Finding a Shortest Path Is as Difficult as
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
the Distance Problem
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Definition
|
2007-07-24 17:45:46 +00:00
|
|
|
Let
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
|
|
|
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
|
|
|
|
|
\color none
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2006-10-31 20:56:42 +00:00
|
|
|
contain all tuples
|
|
|
|
|
\begin_inset Formula $(T,s,t,d)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
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|
|
|
such that
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $T=(V,E)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
is a tournament in which
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
the distance of
|
|
|
|
|
\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
and\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
is at most\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
|
|
|
\begin_inset Formula $d$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
The Tournament Distance Problem is Hard
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Theorem
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
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|
is
|
|
|
|
|
\begin_inset Formula $\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
-complete.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\hfill
|
2006-10-31 20:56:42 +00:00
|
|
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|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hyperlink{hierarchy<6>}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
beamerskipbutton{Skip Proof}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
|
|
|
|
|
|
\begin_layout Corollary
|
2006-10-31 20:56:42 +00:00
|
|
|
Shortest path in tournaments can be constructed
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
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|
in logarithmic space, iff
|
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|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Corollary
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset Formula $\Lang{distance}\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{distance}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
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|
Proof That
|
|
|
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
is NL-complete
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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|
status collapsed
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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|
nointerlineskip
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Columns
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
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|
|
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|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
[t,onlytextwidth]
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Column
|
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|
|
5.7cm
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
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|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
setlength
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
leftmargini{1.5em}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Reduce
|
|
|
|
|
\begin_inset Formula $\Lang{reach}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
to
|
|
|
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<alert@1>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Is input
|
|
|
|
|
\begin_inset Formula $(G,s,t)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
in
|
|
|
|
|
\begin_inset Formula $\Lang{reach}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
?
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<2-| alert@2-8>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Map
|
|
|
|
|
\begin_inset Formula $G$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
to
|
|
|
|
|
\begin_inset Formula $G'$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<9-| alert@9>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Query:
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
?
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Separator
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
Correctness
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<10->
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<10-| alert@10-11>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
A path in\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset Formula $G$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
induces
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
|
|
|
|
a length-3 path in\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset Formula $G'$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<12-| alert@12-13>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
A length-3 path in\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset Formula $G'$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
induces
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
|
|
|
|
a path in\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset Formula $G'$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Column
|
2006-10-31 20:56:42 +00:00
|
|
|
4.5cm
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Example
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
|
|
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color{beamerexample}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfsetlinewidth{0.6pt}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{A}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(1,3.3)}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{B}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(2,3.3)}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{C}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(3,3.3)}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{D}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(4,3.3)}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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color{white}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfputat{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodecenter{A}}{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfbox[center,center]{$s$}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodecenter{D}}{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfbox[center,center]{$t$}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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color{beamerexample}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfsetendarrow{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfarrowto}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodesetsepstart{2pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodesetsepend{2pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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alert<3>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B}{A}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<4>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B}{C}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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alert<5,10-11,13>{
|
2006-11-01 08:41:31 +00:00
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodeconnline{C}{D}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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alert<6,10-11,13>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodeconncurve{A}{C}{45}{135}{15pt}{15pt}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfxy(0,3.3)}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfbox[left,center]{$G
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
colon$}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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only<2->{
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(0,2.25)}{
|
2006-11-01 08:41:31 +00:00
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfbox[left,center]{$G'
|
2006-11-01 08:41:31 +00:00
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|
\backslash
|
2006-10-31 20:56:42 +00:00
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colon$}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{A1}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(1,2.25)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{B1}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfxy(2,2.25)}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{C1}{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfxy(3,2.25)}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
graphnode{D1}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfxy(4,2.25)}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{A2}{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(1,1.25)}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{B2}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(2,1.25)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{C2}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(3,1.25)}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{D2}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(4,1.25)}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{A3}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(1,0.25)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{B3}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfxy(2,0.25)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{C3}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(3,0.25)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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graphnode{D3}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(4,0.25)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{A4}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(1,-.75)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{B4}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(2,-.75)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{C4}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(3,-.75)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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graphnode{D4}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfxy(4,-.75)}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
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{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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color{white}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfnodecenter{A1}}{
|
2006-11-01 08:41:31 +00:00
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfbox[center,center]{$s'$}}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfputat{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodecenter{D4}}{
|
2006-11-01 08:41:31 +00:00
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfbox[center,center]{$t'$}}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
|
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}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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only<8->{
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetlinewidth{0.4pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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color{beamerexample!25!averagebackgroundcolor}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A2}{C1}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A2}{D1}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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|
\end_layout
|
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B2}{A1}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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\begin_layout Standard
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B2}{C1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodeconnline{B2}{D1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
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|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C2}{D1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
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|
\end_layout
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D2}{A1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
|
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|
\begin_layout Standard
|
|
|
|
|
|
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|
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|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D2}{B1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A3}{C2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
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|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A3}{D2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
|
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|
|
|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B3}{A2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B3}{C2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
|
|
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|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B3}{D2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
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|
\begin_layout Standard
|
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|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C3}{D2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D3}{A2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D3}{B2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A4}{C3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A4}{D3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B4}{A3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B4}{C3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B4}{D3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C4}{D3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D4}{A3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{D4}{B3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfsetstartarrow{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfarrowto}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A1}{B1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B1}{C1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C1}{D1}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A2}{B2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B2}{C2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C2}{D2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A3}{B3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B3}{C3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C3}{D3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{A4}{B4}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B4}{C4}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{C4}{D4}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfclearstartarrow
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconncurve{B4}{B2}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconncurve{B4}{B1}{135}{-135}{15pt}{15pt}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodeconncurve{C3}{C1}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
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2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodeconncurve{C4}{C2}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
pgfnodeconncurve{C4}{C1}{135}{-135}{15pt}{15pt}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
|
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconncurve{D3}{D1}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconncurve{D4}{D2}{135}{-135}{10pt}{10pt}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
|
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\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconncurve{D4}{D1}{135}{-135}{15pt}{15pt}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
color{beamerexample}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfsetlinewidth{0.6pt}
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
|
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\end_layout
|
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\begin_layout Standard
|
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\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<3->{
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color<3>{red}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B1}{A2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B2}{A3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B3}{A4}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
|
|
}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<4->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color<4>{red}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B1}{C2}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
\end_layout
|
|
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|
\begin_layout Standard
|
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|
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|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B2}{C3}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
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|
\begin_layout Standard
|
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|
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|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
pgfnodeconnline{B3}{C4}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
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|
\begin_layout Standard
|
|
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|
|
|
|
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|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
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|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
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|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
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|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
only<5->{
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
color<5>{red}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
|
|
|
|
|
\end_layout
|
|
|
|
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C1}{D2}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<11>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C2}{D3}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<12-13>{
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C3}{D4}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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only<6->{
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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color<6>{red}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<11>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A1}{C2}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert<12-13>{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A2}{C3}}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A3}{C4}
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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only<7->{
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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color<7>{red}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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alert<12-13>{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodeconnline{A1}{A2}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A2}{A3}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{A3}{A4}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B1}{B2}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B2}{B3}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{B3}{B4}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C1}{C2}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C2}{C3}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{C3}{C4}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{D1}{D2}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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pgfnodeconnline{D2}{D3}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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2006-11-01 08:41:31 +00:00
|
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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alert<11>{
|
2006-11-01 08:41:31 +00:00
|
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|
\backslash
|
2006-10-31 20:56:42 +00:00
|
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pgfnodeconnline{D3}{D4}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Standard
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\end_layout
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
|
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}
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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\end_layout
|
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\begin_layout Standard
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\backslash
|
2006-10-31 20:56:42 +00:00
|
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|
end{pgfpicture}
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
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\end_inset
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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|
\end_deeper
|
|
|
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|
\begin_layout AgainFrame
|
2006-10-31 20:56:42 +00:00
|
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|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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|
<6>
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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|
hierarchy
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Subsection
|
2006-10-31 20:56:42 +00:00
|
|
|
Complexity of: Approximating the Shortest Path
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Definition
|
2007-07-24 17:45:46 +00:00
|
|
|
An
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
approximation scheme for
|
|
|
|
|
\begin_inset Formula $\Lang{tournament-shortest-path}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
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|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
|
|
|
|
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2007-07-24 17:45:46 +00:00
|
|
|
gets as input
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
a tuple
|
|
|
|
|
\begin_inset Formula $(T,s,t)\in\Lang{reach}_{\operatorname{tourn}}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
and
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
a number
|
|
|
|
|
\begin_inset Formula $r>1$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
It outputs
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
a path from
|
|
|
|
|
\begin_inset Formula $s$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
to\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset Formula $t$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
of length at most
|
|
|
|
|
\begin_inset Formula $r\operatorname{d_{T}}(s,t)$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
There Exists a Logspace Approximation Scheme for
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
|
|
|
|
the Tournament Shortest
|
|
|
|
|
Path Problem
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Theorem
|
2006-10-31 20:56:42 +00:00
|
|
|
There exists an approximation scheme for
|
|
|
|
|
\begin_inset Formula $\Lang{tournament-shortest-path}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
that for
|
|
|
|
|
\begin_inset Formula $1<r<2$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
needs space
|
|
|
|
|
\begin_inset Formula \[
|
|
|
|
|
O\left(\log|V|\log\frac{1}{r-1}\right).\]
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Pause
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Corollary
|
2006-10-31 20:56:42 +00:00
|
|
|
In tournaments, paths can be constructed in logarithmic space.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\hfill
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hyperlink{optimality}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
beamergotobutton{More Details}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout AgainFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status collapsed
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
<7>
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
hierarchy
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Section*
|
2006-10-31 20:56:42 +00:00
|
|
|
Summary
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Subsection*
|
2006-10-31 20:56:42 +00:00
|
|
|
Summary
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
|
|
|
Summary
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{Summary}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2007-07-24 17:45:46 +00:00
|
|
|
Tournament
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
reachability
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
|
|
|
|
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2007-07-24 17:45:46 +00:00
|
|
|
is in
|
|
|
|
|
\color none
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
\color red
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Class{AC}^{0}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2006-10-31 20:56:42 +00:00
|
|
|
.
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2007-07-24 17:45:46 +00:00
|
|
|
There exists a
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
logspace approximation scheme
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
|
|
|
|
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2007-07-24 17:45:46 +00:00
|
|
|
for
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
approximating
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
|
|
|
|
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2007-07-24 17:45:46 +00:00
|
|
|
shortest paths in tournaments.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Itemize
|
2007-07-24 17:45:46 +00:00
|
|
|
Finding
|
|
|
|
|
\color none
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\color red
|
|
|
|
|
shortest paths
|
2007-07-24 17:45:46 +00:00
|
|
|
\color none
|
|
|
|
|
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2007-07-24 17:45:46 +00:00
|
|
|
in tournaments is
|
|
|
|
|
\color none
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2007-07-24 17:45:46 +00:00
|
|
|
\color red
|
|
|
|
|
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset Formula $\Class{NL}$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
-complete
|
2006-11-01 09:22:35 +00:00
|
|
|
\color inherit
|
2006-10-31 20:56:42 +00:00
|
|
|
.
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Separator
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Block
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
{Outlook}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_deeper
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
The same results apply to graphs with
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
bounded independence number.
|
2006-11-01 08:41:31 +00:00
|
|
|
\hfill
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hyperlink{independence}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
beamergotobutton{More Details}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
|
|
The complexity of finding paths in undirected graphs
|
2006-11-01 08:41:31 +00:00
|
|
|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
is partly open.
|
2006-11-01 08:41:31 +00:00
|
|
|
\hfill
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
hyperlink{undirected}{
|
2006-11-01 08:41:31 +00:00
|
|
|
\backslash
|
2006-10-31 20:56:42 +00:00
|
|
|
beamergotobutton{More Details}}
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
|
|
|
|
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_deeper
|
|
|
|
|
\begin_layout Subsection*
|
2006-10-31 20:56:42 +00:00
|
|
|
For Further Reading
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout BeginFrame
|
|
|
|
|
For Further Reading
|
|
|
|
|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
|
|
|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
|
|
\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
|
|
status inlined
|
2006-10-31 20:56:42 +00:00
|
|
|
|
2006-11-01 08:41:31 +00:00
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key "Moon1968"
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2006-10-31 20:56:42 +00:00
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John Moon.
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\emph on
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2006-10-31 20:56:42 +00:00
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Topics on Tournaments.
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Holt, Rinehart, and Winston, 1968.
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\begin_layout Bibliography
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2007-07-24 17:45:46 +00:00
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\begin_inset LatexCommand bibitem
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key "NickelsenT2002"
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2006-11-01 08:41:31 +00:00
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2006-10-31 20:56:42 +00:00
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Arfst Nickelsen and Till Tantau.
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2006-11-01 08:41:31 +00:00
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2006-11-01 08:41:31 +00:00
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2006-10-31 20:56:42 +00:00
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On reachability in graphs with bounded independence number.
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2006-11-01 08:41:31 +00:00
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status collapsed
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2006-10-31 20:56:42 +00:00
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In
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\emph default
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2006-11-01 08:41:31 +00:00
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\emph on
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2006-10-31 20:56:42 +00:00
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Proc.
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of COCOON 2002
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2006-11-01 08:41:31 +00:00
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\emph default
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2006-10-31 20:56:42 +00:00
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, Springer-Verlag, 2002.
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Bibliography
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2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
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\begin_inset LatexCommand bibitem
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key "Tantau2004b"
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\end_inset
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2006-11-01 08:41:31 +00:00
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\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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Till Tantau
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2006-11-01 08:41:31 +00:00
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status collapsed
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2006-10-31 20:56:42 +00:00
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A logspace approximation scheme for the shortest path problem for graphs
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with bounded independence number.
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\begin_inset ERT
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2006-11-01 08:41:31 +00:00
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In
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\emph default
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Proc.
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of STACS 2004
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2006-11-01 08:41:31 +00:00
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\emph default
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2006-10-31 20:56:42 +00:00
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, Springer-Verlag, 2004.
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In press.
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\start_of_appendix
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\backslash
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AtBeginSubsection[]{}
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\begin_layout Section
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Appendix
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2006-11-01 08:41:31 +00:00
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\begin_layout Subsection
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Graphs With Bounded Independence Number
|
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\end_layout
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\begin_layout BeginFrame
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[label=independence]
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Definition of Independence Number of a Graph
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\end_layout
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\begin_layout Definition
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The
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\color none
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\color red
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independence number
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\color none
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\color inherit
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\begin_inset Formula $\alpha(G)$
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of a directed graph
|
2006-11-01 08:41:31 +00:00
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\newline
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is the maximum number of vertices we can pick,
|
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\newline
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such that
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there is no edge between them.
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\begin_layout Example
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Tournaments have independence number 1.
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\end_layout
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2006-11-01 08:41:31 +00:00
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\begin_layout BeginFrame
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The Results for Tournaments also Apply to
|
2006-11-01 08:41:31 +00:00
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\newline
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Graphs With Bounded Independence
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Number
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\end_layout
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2006-11-01 08:41:31 +00:00
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\begin_layout Theorem
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For each\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $k$
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\end_inset
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,
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\color none
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2006-10-31 20:56:42 +00:00
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\color red
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reachability
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2007-07-24 17:45:46 +00:00
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\color none
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2006-11-01 09:22:35 +00:00
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\color inherit
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2007-07-24 17:45:46 +00:00
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in graphs with independence number
|
2006-11-01 08:41:31 +00:00
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\newline
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at most\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $k$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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is in
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\begin_inset Formula $\Class{AC}^{0}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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.
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Separator
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2006-11-01 08:41:31 +00:00
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\end_layout
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\begin_layout Theorem
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For each\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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\begin_inset Formula $k$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
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, there exists a
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\color none
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2006-10-31 20:56:42 +00:00
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\color red
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logspace approximation scheme
|
2007-07-24 17:45:46 +00:00
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\color none
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2006-11-01 09:22:35 +00:00
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\color inherit
|
2007-07-24 17:45:46 +00:00
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for approximating the shortest path in graphs with independence number
|
2006-11-01 08:41:31 +00:00
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at most\InsetSpace ~
|
2006-10-31 20:56:42 +00:00
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\begin_inset Formula $k$
|
2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Separator
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Theorem
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For each\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $k$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
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, finding the
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\color none
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2006-10-31 20:56:42 +00:00
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\color red
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shortest path
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2007-07-24 17:45:46 +00:00
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\color none
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2006-11-01 09:22:35 +00:00
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\color inherit
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2007-07-24 17:45:46 +00:00
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in graphs with independence number at most\InsetSpace ~
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $k$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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2007-07-24 17:45:46 +00:00
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is
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\color none
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2006-10-31 20:56:42 +00:00
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\color red
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\begin_inset Formula $\Class{NL}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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-complete
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2006-11-01 09:22:35 +00:00
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\color inherit
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2006-10-31 20:56:42 +00:00
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.
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Subsection
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2006-10-31 20:56:42 +00:00
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Finding Paths in Undirected Graphs
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout BeginFrame
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2006-10-31 20:56:42 +00:00
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\begin_inset ERT
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2006-11-01 08:41:31 +00:00
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status inlined
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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<1-2>[label=undirected]
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2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
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2006-10-31 20:56:42 +00:00
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The Complexity of Finding Paths in Undirected Graphs
|
2006-11-01 08:41:31 +00:00
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\newline
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2006-10-31 20:56:42 +00:00
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Is Party Unknown.
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Fact
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2006-10-31 20:56:42 +00:00
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\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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is
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\begin_inset Formula $\Class{SL}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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-complete.
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Corollary
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2006-10-31 20:56:42 +00:00
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For undirected graphs, we can solve
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_deeper
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\begin_layout Itemize
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2006-10-31 20:56:42 +00:00
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the reachability problem in logspace iff
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\begin_inset Formula $\Class L=\Class{SL}$
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2006-11-01 08:41:31 +00:00
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\end_inset
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2006-10-31 20:56:42 +00:00
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,
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Itemize
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2006-10-31 20:56:42 +00:00
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the construction problem in logspace iff
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alt<1>{?}{
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alert{$
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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Class L =
|
2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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Class{SL}$}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
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2006-10-31 20:56:42 +00:00
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,
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
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the optimization problem in logspace iff
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status inlined
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Standard
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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alt<1>{?}{
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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alert{$
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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Class L =
|
2006-11-01 08:41:31 +00:00
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\backslash
|
2006-10-31 20:56:42 +00:00
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Class{NL}$}}
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
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,
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Itemize
|
2006-10-31 20:56:42 +00:00
|
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|
the approximation problem in logspace iff ?.
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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|
\end_deeper
|
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\begin_layout Subsection
|
2006-10-31 20:56:42 +00:00
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The Approximation Scheme is Optimal
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout BeginFrame
|
2006-10-31 20:56:42 +00:00
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
|
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|
status inlined
|
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|
\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
|
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[label=optimality]
|
2006-11-01 08:41:31 +00:00
|
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|
\end_layout
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\end_inset
|
2006-10-31 20:56:42 +00:00
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|
The Approximation Scheme is Optimal
|
2006-11-01 08:41:31 +00:00
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|
\end_layout
|
2006-10-31 20:56:42 +00:00
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|
2006-11-01 08:41:31 +00:00
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|
\begin_layout Theorem
|
2006-10-31 20:56:42 +00:00
|
|
|
Suppose there exists an approximation scheme for
|
|
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|
\begin_inset Formula $\Lang{tournament-shortest-path}$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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that needs space
|
|
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|
\begin_inset Formula $O\bigl(\log|V|\log^{1-\epsilon}\frac{1}{r-1}\bigr)$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
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.
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|
Then
|
|
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|
\begin_inset Formula $\Class{NL}\subseteq\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
|
2006-11-01 08:41:31 +00:00
|
|
|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
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|
\begin_layout Proof
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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|
\begin_deeper
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|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
Suppose the approximation scheme exists.
|
2006-11-01 08:41:31 +00:00
|
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|
\newline
|
2006-10-31 20:56:42 +00:00
|
|
|
We show
|
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|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
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.
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2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
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|
Let
|
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|
\begin_inset Formula $(T,s,t)$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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be an input.
|
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|
Let
|
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|
\begin_inset Formula $n$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
|
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be the number of vertices.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
|
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|
2006-11-01 08:41:31 +00:00
|
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|
\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
|
|
|
Run the approximation scheme for
|
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|
\begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$
|
2006-11-01 08:41:31 +00:00
|
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|
\end_inset
|
2006-10-31 20:56:42 +00:00
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.
|
2006-11-01 08:41:31 +00:00
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\newline
|
2006-10-31 20:56:42 +00:00
|
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This needs space
|
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\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
|
2006-11-01 08:41:31 +00:00
|
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\end_inset
|
2006-10-31 20:56:42 +00:00
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.
|
2006-11-01 08:41:31 +00:00
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\begin_layout Enumerate
|
2006-10-31 20:56:42 +00:00
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|
The resulting path has optimal length.
|
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\begin_inset ERT
|
2006-11-01 08:41:31 +00:00
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status collapsed
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
|
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\begin_layout Standard
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\backslash
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2006-10-31 20:56:42 +00:00
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qedhere
|
2006-11-01 08:41:31 +00:00
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\end_layout
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\end_inset
|
|
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\end_layout
|
2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_deeper
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\begin_layout EndFrame
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_layout
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2006-10-31 20:56:42 +00:00
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2006-11-01 08:41:31 +00:00
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\end_body
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\end_document
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