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The difference has been validated with diffpdf. The following could not be tested and were left unchanged: docbook_article.lyx lilypond.lyx linguistics.lyx springer/sv*.lyx de/linguistics.lyx es/linguistics.lyx ja/FeynmanDiagrams.lyx ja/lilypond.lyx ja/beamer.lyx ja/xypic.lyx The following is the script that I used (in lib/examples): LYX=../../build/src/lyx $LYX -E pdf2 $1.old.pdf $1 sed -i "s/^\\\\begin_inset Separator parbreak$/\\\\begin_inset Separator plain/" $1 sed -i "/^\\\\begin_inset Separator latexpar$/ { N; d; }" $1 $LYX -e lyx $1 $LYX -E pdf2 $1.pdf $1 diffpdf $1.old.pdf $1.pdf
6107 lines
73 KiB
Plaintext
6107 lines
73 KiB
Plaintext
#LyX 2.2 created this file. For more info see http://www.lyx.org/
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\lyxformat 508
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\begin_document
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\begin_header
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\save_transient_properties true
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\origin /systemlyxdir/examples/
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\textclass beamer
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\begin_preamble
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\beamertemplateshadingbackground{red!5}{structure!5}
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\usepackage{beamerthemeshadow}
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\usepackage{pgfnodes,pgfarrows,pgfheaps}
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\beamertemplatetransparentcovereddynamicmedium
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\logo{\pgfuseimage{icsi-logo}}
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\newcommand{\Class}[1]{\operatorname{\mathchoice
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{\text{\small #1}}
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{\text{\small #1}}
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{\text{#1}}
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{\text{#1}}}}
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\newcommand{\Lang}[1]{\operatorname{\text{\textsc{#1}}}}
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% This gets defined by beamerbasecolor.sty, but only at the beginning of
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% the document
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\colorlet{averagebackgroundcolor}{normal text.bg}
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\newcommand{\tape}[3]{%
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\color{structure!30!averagebackgroundcolor}
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}
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}
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{color(0pt)=(black); color(1cm)=(red!35!white)}
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\newcommand{\heap}[5]{%
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\begin{pgfscope}
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\color{#4}
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\pgfheappath{\pgfxy(0,#1)}{\pgfxy(-#2,0)}{\pgfxy(#2,0)}
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\end{pgfmagnify}
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\end{pgfscope}
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%\pgffill
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\color{#4}
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\pgfstroke
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\color{white}
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\pgfheaplabel{\pgfxy(0,#1)}{#3}%
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}
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\newcommand{\langat}[2]{%
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\color{black!30!beamerexample}
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\pgfsetlinewidth{0.6pt}
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\pgfsetendarrow{\pgfarrowdot}
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\pgfline{\pgfxy(-3.5,#1)}{\pgfxy(0.05,#1)}
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\color{beamerexample}
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\pgfputat{\pgfxy(-3.6,#1)}{\pgfbox[right,center]{#2}}%
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}
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\newcommand{\langatother}[2]{%
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\pgfputat{\pgfxy(3.6,#1)}{\pgfbox[left,center]{#2}}%
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}
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\pgfdeclaremask{knight1-mask}{beamer-knight1-mask} \pgfdeclareimage[height=2cm,mask=knight1-mask]{knight1}{beamer-knight1} \pgfdeclaremask{knight2-mask}{beamer-knight2-mask} \pgfdeclareimage[height=2cm,mask=knight2-mask]{knight2}{beamer-knight2} \pgfdeclaremask{knight3-mask}{beamer-knight3-mask} \pgfdeclareimage[height=2cm,mask=knight3-mask,interpolate=true]{knight3}{beamer-knight3} \pgfdeclaremask{knight4-mask}{beamer-knight4-mask} \pgfdeclareimage[height=2cm,mask=knight4-mask,interpolate=true]{knight4}{beamer-knight4}
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\pgfdeclareradialshading{graphnode}
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{\pgfpoint{-3pt}{3.6pt}}%
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{color(0cm)=(beamerexample!15);
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color(2.63pt)=(beamerexample!75);
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color(5.26pt)=(beamerexample!70!black);
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color(7.6pt)=(beamerexample!50!black);
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color(8pt)=(beamerexample!10!averagebackgroundcolor)}
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\newcommand{\graphnode}[2]{
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\pgfnodecircle{#1}[virtual]{#2}{8pt}
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\pgfputat{#2}{\pgfbox[center,center]{\pgfuseshading{graphnode}}}
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}
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\end_preamble
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\options notes=show
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\use_default_options false
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\maintain_unincluded_children false
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\language english
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\language_package default
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\inputencoding auto
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\fontencoding global
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\font_roman "times" "default"
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\font_sans "default" "default"
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\font_typewriter "default" "default"
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\font_math "auto" "auto"
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\font_default_family default
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\use_non_tex_fonts false
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\font_sc false
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\font_osf false
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\font_sf_scale 100 100
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\font_tt_scale 100 100
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\graphics default
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\default_output_format default
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\output_sync 0
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\bibtex_command default
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\index_command default
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\paperfontsize default
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\spacing single
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\use_hyperref false
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\papersize default
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\use_geometry false
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\use_package amsmath 2
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\use_package amssymb 2
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\use_package cancel 0
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\use_package esint 0
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\use_package mathdots 1
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\use_package mathtools 0
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\use_package mhchem 1
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\use_package stackrel 0
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\use_package stmaryrd 0
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\use_package undertilde 0
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\cite_engine basic
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\cite_engine_type default
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\biblio_style plain
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\use_bibtopic false
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\use_indices false
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\paperorientation portrait
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\suppress_date false
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\justification true
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\use_refstyle 0
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 2
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\tocdepth 2
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\paragraph_separation indent
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\paragraph_indentation default
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\quotes_language english
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\papercolumns 1
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\papersides 1
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\paperpagestyle default
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\tracking_changes false
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\output_changes false
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\html_css_as_file 0
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\html_be_strict 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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\begin_inset Newline newline
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\end_inset
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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 Science Institute
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\begin_inset Newline newline
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\end_inset
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Berkeley, California
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\begin_inset Argument 1
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status collapsed
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\begin_layout Plain Layout
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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 Frame
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\begin_inset Argument 4
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status open
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\begin_layout Plain Layout
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Outline
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\end_layout
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\end_inset
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\end_layout
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\begin_deeper
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\begin_layout Standard
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\begin_inset CommandInset toc
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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 Plain Layout
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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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\end_deeper
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\begin_layout Standard
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\begin_inset ERT
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status open
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\begin_layout Plain Layout
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% Show the table of contents at the beginning
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\end_layout
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\begin_layout Plain Layout
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% of every subsection.
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\end_layout
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\begin_layout Plain Layout
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\backslash
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AtBeginSubsection[]{%
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\end_layout
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\begin_layout Plain 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 Plain Layout
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\backslash
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frametitle{Outline}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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tableofcontents[current,currentsubsection]
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\end_layout
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\begin_layout Plain Layout
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}
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\end_layout
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\begin_layout Plain Layout
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}
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Section
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Introduction
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\end_layout
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\begin_layout Subsection
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What are Tournaments?
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\end_layout
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\begin_layout Frame
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\begin_inset Argument 4
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status open
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\begin_layout Plain Layout
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Tournaments Consist of Jousts Between Knights
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Columns
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\begin_layout Column
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5.75cm
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status collapsed
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\backslash
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begin{pgfpicture}{1.25cm}{-1cm}{7cm}{4cm}
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pgfnodebox{A}[virtual]{
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pgfxy(2,1)}{%
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\backslash
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pgfuseimage{knight1}}{2pt}{2pt}
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pgfnodebox{B}[virtual]{
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pgfxy(6,1)}{%
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pgfuseimage{knight2}}{2pt}{2pt}
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pgfnodebox{C}[virtual]{
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\backslash
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pgfxy(4,-1)}{%
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\backslash
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pgfuseimage{knight3}}{2pt}{2pt}
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pgfnodebox{D}[virtual]{
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\backslash
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pgfxy(4,3)}{%
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\backslash
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pgfuseimage{knight4}}{2pt}{2pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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color{beamerexample}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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only<3->{
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\backslash
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pgfsetendarrow{
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\backslash
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pgfarrowto}}
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\begin_layout Plain Layout
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\backslash
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only<2->{%
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfsetlinewidth{0.6pt}
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\end_layout
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\backslash
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pgfnodeconnline{A}{B}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconnline{A}{C}
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\end_layout
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\backslash
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pgfnodeconnline{D}{A}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconnline{C}{B}
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\end_layout
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\backslash
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pgfnodeconnline{B}{D}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconnline{C}{D}
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\end_layout
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\begin_layout Plain Layout
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}
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\begin_layout Plain Layout
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\backslash
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end{pgfpicture}
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Column
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6cm
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\end_layout
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\begin_layout Block
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\begin_inset Argument 2
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status open
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\begin_layout Plain Layout
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What is a Tournament?
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\end_layout
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\end_inset
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\end_layout
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\begin_deeper
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\begin_layout Itemize
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\begin_inset Argument item:2
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status open
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\begin_layout Plain Layout
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1-
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\end_layout
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\end_inset
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A group of knights.
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\end_layout
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\begin_layout Itemize
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\begin_inset Argument item:2
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status open
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\begin_layout Plain Layout
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2-
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\end_layout
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\end_inset
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Every pair has a joust.
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\end_layout
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\begin_layout Itemize
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\begin_inset Argument item:2
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status open
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\begin_layout Plain Layout
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3-
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\end_layout
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\end_inset
|
|
|
|
In every joust one knight wins.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Tournaments are Complete Directed Graphs
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Columns
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Column
|
|
5cm
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
begin{pgfpicture}{1.5cm}{-1cm}{6.5cm}{4cm}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{0.6pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A}{
|
|
\backslash
|
|
pgfxy(2.5,1)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B}{
|
|
\backslash
|
|
pgfxy(5.5,1)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C}{
|
|
\backslash
|
|
pgfxy(4,-0.5)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D}{
|
|
\backslash
|
|
pgfxy(4,2.5)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{white}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{A}}{
|
|
\backslash
|
|
pgfbox[center,center]{$v_2$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{B}}{
|
|
\backslash
|
|
pgfbox[center,center]{$v_3$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{C}}{
|
|
\backslash
|
|
pgfbox[center,center]{$v_4$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{D}}{
|
|
\backslash
|
|
pgfbox[center,center]{$v_1$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetendarrow{
|
|
\backslash
|
|
pgfarrowto}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepstart{2pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepend{4pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A}{B}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A}{C}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D}{A}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C}{B}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B}{D}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D}{C}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
end{pgfpicture}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Column
|
|
6cm
|
|
\end_layout
|
|
|
|
\begin_layout Definition
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
2-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
A
|
|
\color none
|
|
|
|
\color red
|
|
tournament
|
|
\color none
|
|
|
|
\color inherit
|
|
is a
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Enumerate
|
|
directed graphs,
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
with exactly one edge between
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
any two different vertices.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
+
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Tournaments Arise Naturally in Different Situations
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout ExampleBlock
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Applications in Ordering Theory
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
Elements in a set need to be sorted.
|
|
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
The comparison relation may be cyclic, however.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout ExampleBlock
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Applications in Sociology
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
Several candidates apply for a position.
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
Reviewers decide for any two candidates whom they prefer.
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout ExampleBlock
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Applications in Structural Complexity Theory
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
A language
|
|
\begin_inset Formula $L$
|
|
\end_inset
|
|
|
|
is given and a selector function
|
|
\begin_inset Formula $f$
|
|
\end_inset
|
|
|
|
.
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
It chooses from any two words the one more likely to be in
|
|
\begin_inset Formula $f$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Subsection
|
|
What Does ``Finding Paths'' Mean?
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
``Finding Paths'' is Ambiguous
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Input for
|
|
\begin_inset Flex Only
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Path Finding Problems
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Flex Only
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
2-3
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $\Lang{reach}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Flex Only
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
4-5
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
the Construction Problem
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Flex Only
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
6-7
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
the Optimization Problem
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Flex Only
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
8-9
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $\Lang{distance}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Flex Only
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
10-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
the Approximation Problem
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
A
|
|
\color none
|
|
|
|
\color red
|
|
graph
|
|
\color none
|
|
|
|
\color inherit
|
|
|
|
\begin_inset Formula $G=(V,E)$
|
|
\end_inset
|
|
|
|
, a
|
|
\color none
|
|
|
|
\color red
|
|
source
|
|
\color none
|
|
|
|
\color inherit
|
|
|
|
\begin_inset Formula $s\in V$
|
|
\end_inset
|
|
|
|
and a
|
|
\color none
|
|
|
|
\color red
|
|
target
|
|
\color none
|
|
|
|
\color inherit
|
|
|
|
\begin_inset Formula $t\in V$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Argument item:2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
only@-9| visible@8-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
A
|
|
\color none
|
|
|
|
\color red
|
|
maximum distance
|
|
\color inherit
|
|
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
.
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
phantom{p}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Argument item:2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
only@10-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
An
|
|
\color none
|
|
|
|
\color red
|
|
approximation ratio
|
|
\color none
|
|
|
|
\color inherit
|
|
|
|
\begin_inset Formula $r>1$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
nointerlineskip
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Overprint
|
|
\begin_inset Argument item:1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
1,3,5,7,9,11-12
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Columns
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
t,onlytextwidth
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Flex Alternative
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
1-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
column{
|
|
\backslash
|
|
textwidth}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
column{5cm}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout ExampleBlock
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Example Input
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{0.6pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A}{
|
|
\backslash
|
|
pgfxy(3,1)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B}{
|
|
\backslash
|
|
pgfxy(5,1)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C}{
|
|
\backslash
|
|
pgfxy(4,0)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D}{
|
|
\backslash
|
|
pgfxy(4,2)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{white}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{B}}{
|
|
\backslash
|
|
pgfbox[center,center]{$t$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{D}}{
|
|
\backslash
|
|
pgfbox[center,center]{$s$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetendarrow{
|
|
\backslash
|
|
pgfarrowto}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepstart{2pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepend{4pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A}{B}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A}{C}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D}{A}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C}{B}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B}{D}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D}{C}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<9> {
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(5.3,1)}{
|
|
\backslash
|
|
pgfbox[left,center]{, $d=2$}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<11>{
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(5.3,1)}{
|
|
\backslash
|
|
pgfbox[left,center]{, $r=1.5$}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<12>{
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(5.3,1)}{
|
|
\backslash
|
|
pgfbox[left,center]{, $r=1.25$}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
end{pgfpicture}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Flex Only
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
3-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
column{5cm}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout ExampleBlock
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
only@3-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Example Output
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
begin{pgfpicture}{2.5cm}{-0.6cm}{7.5cm}{2.6cm}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<5-8,10->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{0.6pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A}{
|
|
\backslash
|
|
pgfxy(3,1)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B}{
|
|
\backslash
|
|
pgfxy(5,1)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C}{
|
|
\backslash
|
|
pgfxy(4,0)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D}{
|
|
\backslash
|
|
pgfxy(4,2)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{white}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{B}}{
|
|
\backslash
|
|
pgfbox[center,center]{$t$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{D}}{
|
|
\backslash
|
|
pgfbox[center,center]{$s$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetendarrow{
|
|
\backslash
|
|
pgfarrowto}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepstart{2pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepend{4pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<7,12>{
|
|
\backslash
|
|
pgfnodeconnline{A}{B}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<5,11>{
|
|
\backslash
|
|
pgfnodeconnline{A}{C}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<5,7,11-12>{
|
|
\backslash
|
|
pgfnodeconnline{D}{A}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<5,11>{
|
|
\backslash
|
|
pgfnodeconnline{C}{B}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B}{D}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D}{C}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<3,9>{
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(2.75,1)}{
|
|
\backslash
|
|
pgfbox[left,center]{
|
|
\backslash
|
|
alert{``Yes''}}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
end{pgfpicture}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Overprint
|
|
\begin_inset Argument item:1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
2,4,6,8,10
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Variants of Path Finding Problems
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Description
|
|
\begin_inset Argument item:1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
2-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Reachability
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
Problem: Is there a path from
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
?
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Approximation Problem:
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Description
|
|
\begin_inset Argument item:1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
4-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Construction
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
Problem: Construct a path from
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
?
|
|
\end_layout
|
|
|
|
\begin_layout Description
|
|
\begin_inset Argument item:1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
6-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Optimization
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
Problem: Construct a shortest path from
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Description
|
|
\begin_inset Argument item:1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
8-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Distance
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
Problem: Is the distance of
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
at most
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
?
|
|
\end_layout
|
|
|
|
\begin_layout Description
|
|
\begin_inset Argument item:1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
10-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Approximation
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
Problem: Construct a path from
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
of length
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
approximately their distance.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Section
|
|
Review
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Standard Complexity Classes
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfdeclaremask{computer-mask}{beamer-g4-mask}
|
|
\backslash
|
|
pgfdeclareimage[height=2cm,mask=computer-mask,interpolate=true]{computer}{beamer
|
|
-g4}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Classes L and NL are Defined via
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
Logspace Turing Machines
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
begin{pgfpicture}{-0.5cm}{0cm}{8cm}{5cm}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(0,4)}{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
uncover<2->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(0,0.5)}{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
tape{}{output tape (write only)}{10690836937182}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
uncover<3->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(7,2)}{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
shorttape{work tape (read/write), $O(
|
|
\backslash
|
|
log n)$ symbols}{}{42}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(1.75,2.5)}{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfbox[center,center]{
|
|
\backslash
|
|
pgfuseimage{computer}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{0.6pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{structure}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetendarrow{
|
|
\backslash
|
|
pgfarrowto}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfxycurve(1.75,3.5)(1.75,3.75)(0,3.5)(0,3.85)
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
uncover<2->{
|
|
\backslash
|
|
pgfxycurve(1.75,1.5)(1.75,1)(0,1.5)(0,1.05)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
uncover<3->{
|
|
\backslash
|
|
pgfxycurve(2.65,2.5)(3.75,2.5)(7,1)(7,1.9)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
end{pgfpicture}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Logspace Turing Machines Are Quite Powerful
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Deterministic logspace machines can compute
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
addition, multiplication, and even division
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
reductions used in completeness proofs,
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
reachability in forests.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Non-deterministic logspace machines can compute
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
reachability in graphs,
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
non-reachability in graphs,
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
satisfiability with two literals per clause.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 3
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
label=hierarchy
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Complexity Class Hierarchy
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
begin{pgfpicture}{-5.4cm}{0cm}{5.4cm}{5.5cm}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{0.8pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
heap{5.5}{3.5}{$
|
|
\backslash
|
|
Class P$}{black}{1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetdash{{2pt}}{0pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<2->{
|
|
\backslash
|
|
heap{4.5}{3}{$
|
|
\backslash
|
|
Class{NC}^2$}{black!50!structure}{2}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
heap{3.5}{2.5}{$
|
|
\backslash
|
|
Class{NL}$}{black!50!structure}{3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
heap{2.5}{2}{$
|
|
\backslash
|
|
Class{L}$}{black!50!structure}{4}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<2->{
|
|
\backslash
|
|
heap{1.75}{1.5}{$
|
|
\backslash
|
|
vphantom{A}%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
smash{
|
|
\backslash
|
|
Class{NC}^1}$}{black!50!structure}{5}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetdash{}{0pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<2->{
|
|
\backslash
|
|
heap{1.1}{1}{$
|
|
\backslash
|
|
vphantom{A}%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
smash{
|
|
\backslash
|
|
Class{AC}^0}$}{black}{6}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{1.0pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{black}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfxyline(-5,0)(5,0)
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<1-2>{
|
|
\backslash
|
|
langat{3.375}{$
|
|
\backslash
|
|
Lang{reach}$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<1-2>{
|
|
\backslash
|
|
langat{2.375}{$
|
|
\backslash
|
|
Lang{reach}_{
|
|
\backslash
|
|
operatorname{forest}}$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<2>{
|
|
\backslash
|
|
langat{0.975}{$
|
|
\backslash
|
|
Lang{addition}$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<2>{
|
|
\backslash
|
|
langatother{1.6}{
|
|
\backslash
|
|
vbox{
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{division}$,}
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{parity}$}}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<3-5>{
|
|
\backslash
|
|
langat{3.375}{
|
|
\backslash
|
|
vbox{
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{distance}$,}
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{reach}$}}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<4->{
|
|
\backslash
|
|
langatother{2.375}{
|
|
\backslash
|
|
vbox{
|
|
\backslash
|
|
ignorespaces
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{distance}_{
|
|
\backslash
|
|
operatorname{forest}}$,}
|
|
\backslash
|
|
ignorespaces
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{reach}_{
|
|
\backslash
|
|
operatorname{forest}}$,}
|
|
\backslash
|
|
ignorespaces
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{distance}_{
|
|
\backslash
|
|
operatorname{path}}$,}
|
|
\backslash
|
|
ignorespaces
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{reach}_{
|
|
\backslash
|
|
operatorname{path}}$}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<5->{
|
|
\backslash
|
|
langat{0.975}{$
|
|
\backslash
|
|
Lang{reach}_{
|
|
\backslash
|
|
operatorname{tourn}}$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<6->{
|
|
\backslash
|
|
langat{3.375}{
|
|
\backslash
|
|
vbox{
|
|
\backslash
|
|
ignorespaces
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{distance}_{
|
|
\backslash
|
|
operatorname{tourn}}$,}
|
|
\backslash
|
|
ignorespaces
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{distance}$,}
|
|
\backslash
|
|
ignorespaces
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hbox{$
|
|
\backslash
|
|
Lang{reach}$}}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<7->{
|
|
\backslash
|
|
pgfsetdash{{1pt}}{0pt}%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
langat{2.375}{``$
|
|
\backslash
|
|
Lang{approx}_{
|
|
\backslash
|
|
operatorname{tourn}}$''}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
end{pgfpicture}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Circuit Complexity Classes AC
|
|
\begin_inset Formula $^{0}$
|
|
\end_inset
|
|
|
|
, NC
|
|
\begin_inset Formula $^{1}$
|
|
\end_inset
|
|
|
|
, and NC
|
|
\begin_inset Formula $^{2}$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
Limit the Circuit Depth
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
setlength
|
|
\backslash
|
|
leftmargini{1em}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
nointerlineskip
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Columns
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
t
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Column
|
|
3.6cm
|
|
\end_layout
|
|
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Circuit Class
|
|
\begin_inset Formula $\Class{AC}^{0}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $O(1)$
|
|
\end_inset
|
|
|
|
depth
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
unbounded fan-in
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Examples
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $\Lang{addition}\in\Class{AC}^{0}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $\Lang{parity}\notin\Class{AC}^{0}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Column
|
|
3.6cm
|
|
\end_layout
|
|
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Circuit Class
|
|
\begin_inset Formula $\Class{NC}^{1}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $O(\log n)$
|
|
\end_inset
|
|
|
|
depth
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
bounded fan-in
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Examples
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $\Lang{parity}\in\Class{NC}^{1}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $\Lang{mutiply}\in\Class{NC}^{1}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $\Lang{divide}\in\Class{NC}^{1}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Column
|
|
3.6cm
|
|
\end_layout
|
|
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Circuit Class
|
|
\begin_inset Formula $\Class{NC}^{2}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $O(\log^{2}n)$
|
|
\end_inset
|
|
|
|
depth
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
bounded fan-in
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Examples
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $\Class{NL}\subseteq\Class{NC}^{2}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout AgainFrame
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
hierarchy
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Standard Complexity Results on Finding Paths
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
All Variants of Finding Paths in Directed Graphs
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
Are Equally Difficult
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Fact
|
|
\begin_inset Formula $\Lang{reach}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $\Lang{distance}$
|
|
\end_inset
|
|
|
|
are
|
|
\begin_inset Formula $\Class{NL}$
|
|
\end_inset
|
|
|
|
-complete.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Corollary
|
|
For directed graphs, we can solve
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
the reachability problem in logspace iff
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
the construction problem in logspace iff
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
the optimization problem in logspace iff
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
the approximation problem in logspace iff
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout AgainFrame
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
3
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
hierarchy
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Finding Paths in Forests and Directed Paths is Easy,
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
But Not Trivial
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Fact
|
|
\begin_inset Formula $\Lang{reach}_{\operatorname{forest}}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{forest}}$
|
|
\end_inset
|
|
|
|
are
|
|
\begin_inset Formula $\Class{L}$
|
|
\end_inset
|
|
|
|
-complete.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Fact
|
|
\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{path}}$
|
|
\end_inset
|
|
|
|
are
|
|
\begin_inset Formula $\Class{L}$
|
|
\end_inset
|
|
|
|
-complete.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout AgainFrame
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
4
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
hierarchy
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Finding Paths in Tournaments
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Complexity of: Does a Path Exist?
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Definition of the Tournament Reachability Problem
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Definition
|
|
Let
|
|
\color none
|
|
|
|
\color red
|
|
|
|
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
|
|
\color none
|
|
|
|
\color inherit
|
|
contain all triples
|
|
\begin_inset Formula $(T,s,t)$
|
|
\end_inset
|
|
|
|
such that
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Enumerate
|
|
\begin_inset Formula $T=(V,E)$
|
|
\end_inset
|
|
|
|
is a tournament and
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
there exists a path from
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Tournament Reachability Problem is Very Easy
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Theorem
|
|
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout AlertBlock
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Implications
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
The problem is
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
easier
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
than
|
|
\begin_inset Formula $\Lang{reach}$
|
|
\end_inset
|
|
|
|
and even
|
|
\begin_inset Formula $\Lang{reach}_{\operatorname{path}}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout AgainFrame
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
5
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
hierarchy
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Complexity of: Construct a Shortest Path
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Finding a Shortest Path Is as Difficult as
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
the Distance Problem
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Definition
|
|
Let
|
|
\color none
|
|
|
|
\color red
|
|
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
|
|
\color none
|
|
|
|
\color inherit
|
|
contain all tuples
|
|
\begin_inset Formula $(T,s,t,d)$
|
|
\end_inset
|
|
|
|
such that
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Enumerate
|
|
\begin_inset Formula $T=(V,E)$
|
|
\end_inset
|
|
|
|
is a tournament in which
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
the distance of
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
is at most
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Tournament Distance Problem is Hard
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Theorem
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
is
|
|
\begin_inset Formula $\Class{NL}$
|
|
\end_inset
|
|
|
|
-complete.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset space \hfill{}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hyperlink{hierarchy<6>}{
|
|
\backslash
|
|
beamerskipbutton{Skip Proof}}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Corollary
|
|
Shortest path in tournaments can be constructed
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
in logarithmic space, iff
|
|
\begin_inset Formula $\Class{L}=\Class{NL}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Corollary
|
|
\begin_inset Formula $\Lang{distance}\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{distance}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Proof That
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
is NL-complete
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
nointerlineskip
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Columns
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
t,onlytextwidth
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Column
|
|
5.7cm
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
setlength
|
|
\backslash
|
|
leftmargini{1.5em}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Reduce
|
|
\begin_inset Formula $\Lang{reach}$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Enumerate
|
|
\begin_inset Argument item:2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
alert@1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Is input
|
|
\begin_inset Formula $(G,s,t)$
|
|
\end_inset
|
|
|
|
in
|
|
\begin_inset Formula $\Lang{reach}$
|
|
\end_inset
|
|
|
|
?
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
\begin_inset Argument item:2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
2-| alert@2-8
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Map
|
|
\begin_inset Formula $G$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset Formula $G'$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
\begin_inset Argument item:2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
9-| alert@9
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Query:
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
?
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Correctness
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
10-
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Enumerate
|
|
\begin_inset Argument item:2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
10-| alert@10-11
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
A path in
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $G$
|
|
\end_inset
|
|
|
|
induces
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
a length-3 path in
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $G'$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
\begin_inset Argument item:2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
12-| alert@12-13
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
A length-3 path in
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $G'$
|
|
\end_inset
|
|
|
|
induces
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
a path in
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $G'$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Column
|
|
4.5cm
|
|
\end_layout
|
|
|
|
\begin_layout Example
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
begin{pgfpicture}{0cm}{-1.25cm}{4.5cm}{3.75cm}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{0.6pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A}{
|
|
\backslash
|
|
pgfxy(1,3.3)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B}{
|
|
\backslash
|
|
pgfxy(2,3.3)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C}{
|
|
\backslash
|
|
pgfxy(3,3.3)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D}{
|
|
\backslash
|
|
pgfxy(4,3.3)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{white}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{A}}{
|
|
\backslash
|
|
pgfbox[center,center]{$s$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{D}}{
|
|
\backslash
|
|
pgfbox[center,center]{$t$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetendarrow{
|
|
\backslash
|
|
pgfarrowto}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepstart{2pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodesetsepend{2pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<3>{
|
|
\backslash
|
|
pgfnodeconnline{B}{A}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<4>{
|
|
\backslash
|
|
pgfnodeconnline{B}{C}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<5,10-11,13>{
|
|
\backslash
|
|
pgfnodeconnline{C}{D}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<6,10-11,13>{
|
|
\backslash
|
|
pgfnodeconncurve{A}{C}{45}{135}{15pt}{15pt}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(0,3.3)}{
|
|
\backslash
|
|
pgfbox[left,center]{$G
|
|
\backslash
|
|
colon$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<2->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfxy(0,2.25)}{
|
|
\backslash
|
|
pgfbox[left,center]{$G'
|
|
\backslash
|
|
colon$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A1}{
|
|
\backslash
|
|
pgfxy(1,2.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B1}{
|
|
\backslash
|
|
pgfxy(2,2.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C1}{
|
|
\backslash
|
|
pgfxy(3,2.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D1}{
|
|
\backslash
|
|
pgfxy(4,2.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A2}{
|
|
\backslash
|
|
pgfxy(1,1.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B2}{
|
|
\backslash
|
|
pgfxy(2,1.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C2}{
|
|
\backslash
|
|
pgfxy(3,1.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D2}{
|
|
\backslash
|
|
pgfxy(4,1.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A3}{
|
|
\backslash
|
|
pgfxy(1,0.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B3}{
|
|
\backslash
|
|
pgfxy(2,0.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C3}{
|
|
\backslash
|
|
pgfxy(3,0.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D3}{
|
|
\backslash
|
|
pgfxy(4,0.25)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{A4}{
|
|
\backslash
|
|
pgfxy(1,-.75)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{B4}{
|
|
\backslash
|
|
pgfxy(2,-.75)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{C4}{
|
|
\backslash
|
|
pgfxy(3,-.75)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
graphnode{D4}{
|
|
\backslash
|
|
pgfxy(4,-.75)}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
{
|
|
\backslash
|
|
color{white}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{A1}}{
|
|
\backslash
|
|
pgfbox[center,center]{$s'$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfputat{
|
|
\backslash
|
|
pgfnodecenter{D4}}{
|
|
\backslash
|
|
pgfbox[center,center]{$t'$}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<8->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetlinewidth{0.4pt}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color{beamerexample!25!averagebackgroundcolor}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A2}{C1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A2}{D1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B2}{A1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B2}{C1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B2}{D1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C2}{D1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D2}{A1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D2}{B1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A3}{C2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A3}{D2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B3}{A2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B3}{C2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B3}{D2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C3}{D2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D3}{A2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D3}{B2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A4}{C3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A4}{D3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B4}{A3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B4}{C3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B4}{D3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C4}{D3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D4}{A3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D4}{B3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfsetstartarrow{
|
|
\backslash
|
|
pgfarrowto}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A1}{B1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B1}{C1}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C1}{D1}
|
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
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pgfnodeconnline{A2}{B2}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconnline{B2}{C2}
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
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pgfnodeconnline{C2}{D2}
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
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pgfnodeconnline{A3}{B3}
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
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pgfnodeconnline{B3}{C3}
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
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pgfnodeconnline{C3}{D3}
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
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pgfnodeconnline{A4}{B4}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconnline{B4}{C4}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconnline{C4}{D4}
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
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pgfclearstartarrow
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{B4}{B2}{135}{-135}{10pt}{10pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{B4}{B1}{135}{-135}{15pt}{15pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{C3}{C1}{135}{-135}{10pt}{10pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{C4}{C2}{135}{-135}{10pt}{10pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{C4}{C1}{135}{-135}{15pt}{15pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
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pgfnodeconncurve{D3}{D1}{135}{-135}{10pt}{10pt}
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
|
pgfnodeconncurve{D4}{D2}{135}{-135}{10pt}{10pt}
|
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
|
pgfnodeconncurve{D4}{D1}{135}{-135}{15pt}{15pt}
|
|
\end_layout
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\begin_layout Plain Layout
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\backslash
|
|
color{beamerexample}
|
|
\end_layout
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\begin_layout Plain Layout
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\backslash
|
|
pgfsetlinewidth{0.6pt}
|
|
\end_layout
|
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\begin_layout Plain Layout
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}
|
|
\end_layout
|
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\begin_layout Plain Layout
|
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\backslash
|
|
only<3->{%
|
|
\end_layout
|
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|
\begin_layout Plain Layout
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\backslash
|
|
color<3>{red}
|
|
\end_layout
|
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\begin_layout Plain Layout
|
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\backslash
|
|
pgfnodeconnline{B1}{A2}
|
|
\end_layout
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\begin_layout Plain Layout
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\backslash
|
|
pgfnodeconnline{B2}{A3}
|
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\end_layout
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\begin_layout Plain Layout
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\backslash
|
|
pgfnodeconnline{B3}{A4}
|
|
\end_layout
|
|
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|
\begin_layout Plain Layout
|
|
|
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}
|
|
\end_layout
|
|
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\begin_layout Plain Layout
|
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\backslash
|
|
only<4->{%
|
|
\end_layout
|
|
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|
\begin_layout Plain Layout
|
|
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|
|
\backslash
|
|
color<4>{red}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B1}{C2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B2}{C3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B3}{C4}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<5->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color<5>{red}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C1}{D2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<11>{
|
|
\backslash
|
|
pgfnodeconnline{C2}{D3}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<12-13>{
|
|
\backslash
|
|
pgfnodeconnline{C3}{D4}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<6->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color<6>{red}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<11>{
|
|
\backslash
|
|
pgfnodeconnline{A1}{C2}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<12-13>{
|
|
\backslash
|
|
pgfnodeconnline{A2}{C3}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A3}{C4}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
only<7->{%
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
color<7>{red}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<12-13>{
|
|
\backslash
|
|
pgfnodeconnline{A1}{A2}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A2}{A3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{A3}{A4}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B1}{B2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B2}{B3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{B3}{B4}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C1}{C2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C2}{C3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{C3}{C4}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D1}{D2}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
pgfnodeconnline{D2}{D3}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
alert<11>{
|
|
\backslash
|
|
pgfnodeconnline{D3}{D4}}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
}
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
end{pgfpicture}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout AgainFrame
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
6
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
hierarchy
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Complexity of: Approximating the Shortest Path
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Definition
|
|
An
|
|
\color none
|
|
|
|
\color red
|
|
approximation scheme for
|
|
\begin_inset Formula $\Lang{tournament-shortest-path}$
|
|
\end_inset
|
|
|
|
|
|
\color none
|
|
|
|
\color inherit
|
|
gets as input
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Enumerate
|
|
a tuple
|
|
\begin_inset Formula $(T,s,t)\in\Lang{reach}_{\operatorname{tourn}}$
|
|
\end_inset
|
|
|
|
and
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
a number
|
|
\begin_inset Formula $r>1$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
It outputs
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
a path from
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
of length at most
|
|
\begin_inset Formula $r\operatorname{d_{T}}(s,t)$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
There Exists a Logspace Approximation Scheme for
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
the Tournament Shortest Path Problem
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Theorem
|
|
There exists an approximation scheme for
|
|
\begin_inset Formula $\Lang{tournament-shortest-path}$
|
|
\end_inset
|
|
|
|
that for
|
|
\begin_inset Formula $1<r<2$
|
|
\end_inset
|
|
|
|
needs space
|
|
\begin_inset Formula
|
|
\[
|
|
O\left(\log|V|\log\frac{1}{r-1}\right).
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Pause
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Corollary
|
|
In tournaments, paths can be constructed in logarithmic space.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset space \hfill{}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hyperlink{optimality}{
|
|
\backslash
|
|
beamergotobutton{More Details}}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout AgainFrame
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
7
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
hierarchy
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Note Note
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
If a frame includes a program listing, the frame needs to be marked as
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
fragile
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
.
|
|
\SpecialChar LyX
|
|
has the FragileFrame layout for this.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout FragileFrame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Just a frame with a program code listing
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout FragileFrame
|
|
This is some program code:
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset listings
|
|
lstparams "extendedchars=true,language=Python,numbers=left,stepnumber=3,tabsize=4"
|
|
inline false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
def func(param):
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
'this is a python function'
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
pass
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
def func(param):
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
'This is a German word: Tschüs'
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
pass
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
def func(param):
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
'this is a python function'
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
pass
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Section*
|
|
Summary
|
|
\end_layout
|
|
|
|
\begin_layout Subsection*
|
|
Summary
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Summary
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Summary
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
Tournament
|
|
\color none
|
|
|
|
\color red
|
|
reachability
|
|
\color none
|
|
|
|
\color inherit
|
|
is in
|
|
\color none
|
|
|
|
\color red
|
|
|
|
\begin_inset Formula $\Class{AC}^{0}$
|
|
\end_inset
|
|
|
|
|
|
\color inherit
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
There exists a
|
|
\color none
|
|
|
|
\color red
|
|
logspace approximation scheme
|
|
\color none
|
|
|
|
\color inherit
|
|
for
|
|
\color none
|
|
|
|
\color red
|
|
approximating
|
|
\color none
|
|
|
|
\color inherit
|
|
shortest paths in tournaments.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Finding
|
|
\color none
|
|
|
|
\color red
|
|
shortest paths
|
|
\color none
|
|
|
|
\color inherit
|
|
in tournaments is
|
|
\color none
|
|
|
|
\color red
|
|
|
|
\begin_inset Formula $\Class{NL}$
|
|
\end_inset
|
|
|
|
-complete
|
|
\color inherit
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Block
|
|
\begin_inset Argument 2
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
Outlook
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
The same results apply to graphs with
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
bounded independence number.
|
|
\begin_inset space \hfill{}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hyperlink{independence}{
|
|
\backslash
|
|
beamergotobutton{More Details}}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
The complexity of finding paths in undirected graphs
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
is partly open.
|
|
\begin_inset space \hfill{}
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
hyperlink{undirected}{
|
|
\backslash
|
|
beamergotobutton{More Details}}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Subsection*
|
|
For Further Reading
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
For Further Reading
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
beamertemplatebookbibitems
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "Moon1968"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
John Moon.
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newblock
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\emph on
|
|
Topics on Tournaments.
|
|
|
|
\emph default
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newblock
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
Holt, Rinehart, and Winston, 1968.
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
beamertemplatearticlebibitems
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "NickelsenT2002"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
Arfst Nickelsen and Till Tantau.
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newblock
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
On reachability in graphs with bounded independence number.
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newblock
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
In
|
|
\emph on
|
|
Proc.
|
|
of COCOON 2002
|
|
\emph default
|
|
, Springer-Verlag, 2002.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "Tantau2004b"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
Till Tantau
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newblock
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
A logspace approximation scheme for the shortest path problem for graphs
|
|
with bounded independence number.
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newblock
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
In
|
|
\emph on
|
|
Proc.
|
|
of STACS 2004
|
|
\emph default
|
|
, Springer-Verlag, 2004.
|
|
|
|
\begin_inset ERT
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newblock
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
In press.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\start_of_appendix
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
AtBeginSubsection[]{}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Appendix
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Graphs With Bounded Independence Number
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 3
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
label=independence
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
Definition of Independence Number of a Graph
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Definition
|
|
The
|
|
\color none
|
|
|
|
\color red
|
|
independence number
|
|
\color none
|
|
|
|
\color inherit
|
|
|
|
\begin_inset Formula $\alpha(G)$
|
|
\end_inset
|
|
|
|
of a directed graph
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
is the maximum number of vertices we can pick,
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
such that there is no edge between them.
|
|
\end_layout
|
|
|
|
\begin_layout Example
|
|
Tournaments have independence number 1.
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Results for Tournaments also Apply to
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
Graphs With Bounded Independence Number
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Theorem
|
|
For each
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $k$
|
|
\end_inset
|
|
|
|
,
|
|
\color none
|
|
|
|
\color red
|
|
reachability
|
|
\color none
|
|
|
|
\color inherit
|
|
in graphs with independence number
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
at most
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $k$
|
|
\end_inset
|
|
|
|
is in
|
|
\begin_inset Formula $\Class{AC}^{0}$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Theorem
|
|
For each
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $k$
|
|
\end_inset
|
|
|
|
, there exists a
|
|
\color none
|
|
|
|
\color red
|
|
logspace approximation scheme
|
|
\color none
|
|
|
|
\color inherit
|
|
for approximating the shortest path in graphs with independence number at
|
|
most
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $k$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Separator plain
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Theorem
|
|
For each
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $k$
|
|
\end_inset
|
|
|
|
, finding the
|
|
\color none
|
|
|
|
\color red
|
|
shortest path
|
|
\color none
|
|
|
|
\color inherit
|
|
in graphs with independence number at most
|
|
\begin_inset space ~
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $k$
|
|
\end_inset
|
|
|
|
is
|
|
\color none
|
|
|
|
\color red
|
|
|
|
\begin_inset Formula $\Class{NL}$
|
|
\end_inset
|
|
|
|
-complete
|
|
\color inherit
|
|
.
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\begin_layout Subsection
|
|
Finding Paths in Undirected Graphs
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 1
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
1-2
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 3
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
label=undirected
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Complexity of Finding Paths in Undirected Graphs
|
|
\begin_inset Newline newline
|
|
\end_inset
|
|
|
|
Is Party Unknown.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Fact
|
|
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
|
|
\end_inset
|
|
|
|
is
|
|
\begin_inset Formula $\Class{SL}$
|
|
\end_inset
|
|
|
|
-complete.
|
|
\end_layout
|
|
|
|
\begin_layout Corollary
|
|
For undirected graphs, we can solve
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Itemize
|
|
the reachability problem in logspace iff
|
|
\begin_inset Formula $\Class L=\Class{SL}$
|
|
\end_inset
|
|
|
|
,
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
the construction problem in logspace iff
|
|
\begin_inset Flex Alternative
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
?
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Flex Alert
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\Class L=\Class{SL}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
,
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
the optimization problem in logspace iff
|
|
\begin_inset Flex Alternative
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Argument 1
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
1
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 2
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
?
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Flex Alert
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $\Class L=\Class{NL}$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
,
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
the approximation problem in logspace iff ?.
|
|
|
|
\end_layout
|
|
|
|
\end_deeper
|
|
\end_deeper
|
|
\begin_layout Subsection
|
|
The Approximation Scheme is Optimal
|
|
\end_layout
|
|
|
|
\begin_layout Frame
|
|
\begin_inset Argument 3
|
|
status collapsed
|
|
|
|
\begin_layout Plain Layout
|
|
label=optimality
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Argument 4
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
The Approximation Scheme is Optimal
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
|
|
\begin_layout Theorem
|
|
Suppose there exists an approximation scheme for
|
|
\begin_inset Formula $\Lang{tournament-shortest-path}$
|
|
\end_inset
|
|
|
|
that needs space
|
|
\begin_inset Formula $O\bigl(\log|V|\log^{1-\epsilon}\frac{1}{r-1}\bigr)$
|
|
\end_inset
|
|
|
|
.
|
|
Then
|
|
\begin_inset Formula $\Class{NL}\subseteq\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Proof
|
|
|
|
\end_layout
|
|
|
|
\begin_deeper
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\begin_layout Enumerate
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Suppose the approximation scheme exists.
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\begin_inset Newline newline
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\end_inset
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We show
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\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
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\end_inset
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.
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\end_layout
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\begin_layout Enumerate
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|
Let
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\begin_inset Formula $(T,s,t)$
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\end_inset
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be an input.
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Let
|
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\begin_inset Formula $n$
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\end_inset
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be the number of vertices.
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\end_layout
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\begin_layout Enumerate
|
|
Run the approximation scheme for
|
|
\begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$
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\end_inset
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|
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.
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\begin_inset Newline newline
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\end_inset
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This needs space
|
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\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
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\end_inset
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.
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\end_layout
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\begin_layout Enumerate
|
|
The resulting path has optimal length.
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\begin_inset ERT
|
|
status collapsed
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|
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\begin_layout Plain Layout
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|
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\backslash
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|
qedhere
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\end_layout
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\end_inset
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|
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\end_layout
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\end_deeper
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\end_deeper
|
|
\end_body
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\end_document
|