lyx_mirror/lib/examples/beamerlyxexample1.lyx

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#LyX 2.1 created this file. For more info see http://www.lyx.org/
2012-12-19 18:47:00 +00:00
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2012-12-19 18:47:00 +00:00
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The Complexity of
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Finding Paths in Tournaments
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\begin_layout Author
Till Tantau
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International Computer Science Institute
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Berkeley, California
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ICSI
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January 30th, 2004
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Introduction
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What are Tournaments?
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Tournaments Consist of Jousts Between Knights
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What is a Tournament?
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1-
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A group of knights.
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Every pair has a joust.
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In every joust one knight wins.
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Tournaments are Complete Directed Graphs
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A
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is a
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[<+>]
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Tournaments Arise Naturally in Different Situations
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Applications in Ordering Theory
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Applications in Structural Complexity Theory
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What Does ``Finding Paths'' Mean?
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``Finding Paths'' is Ambiguous
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Path Finding Problems
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the Construction Problem
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\end_deeper
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2,4,6,8,10
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\end_inset
\end_layout
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\begin_deeper
\begin_layout Block
\begin_inset Argument 2
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Variants of Path Finding Problems
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Description
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2-
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\end_inset
Reachability
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Problem: Is there a path from
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\end_inset
to
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\begin_inset Formula $t$
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?
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\begin_inset Argument 2
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Approximation Problem:
\end_layout
\end_inset
\end_layout
\begin_layout Description
\begin_inset Argument item:1
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4-
\end_layout
\end_inset
Construction
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\end_inset
Problem: Construct a path from
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\end_inset
to
\begin_inset space ~
\end_inset
\begin_inset Formula $t$
\end_inset
?
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\begin_layout Description
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6-
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\end_inset
Optimization
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Problem: Construct a shortest path from
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to
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\end_inset
\begin_inset Formula $t$
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.
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8-
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Distance
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and
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\begin_inset Formula $t$
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at most
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\begin_inset Formula $d$
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?
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\begin_layout Description
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10-
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Approximation
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Problem: Construct a path from
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\end_inset
to
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\end_inset
\begin_inset Formula $t$
\end_inset
of length
\begin_inset Newline newline
\end_inset
approximately their distance.
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\end_deeper
\end_deeper
\begin_layout Section
Review
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\begin_layout Subsection
Standard Complexity Classes
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\begin_layout BeginFrame
The Classes L and NL are Defined via
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Logspace Turing Machines
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pgfxy(0,4)}{%
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tape{input tape (read only), $n$ symbols}{}{3401234*3143223=}}
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\begin_layout BeginFrame
Logspace Turing Machines Are Quite Powerful
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\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
\begin_layout BeginFrame
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status collapsed
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<1>[label=hierarchy]
\end_layout
\end_inset
The Complexity Class Hierarchy
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Class P$}{black}{1}
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\begin_layout Plain Layout
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Class{NL}$}{black!50!structure}{3}
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Class{L}$}{black!50!structure}{4}
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}
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\begin_layout Plain Layout
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Class{AC}^0}$}{black}{6}
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}
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\begin_layout Plain Layout
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\backslash
Lang{reach}$}}
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\begin_layout Plain Layout
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\backslash
Lang{reach}_{
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Lang{distance}_{
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ignorespaces
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Lang{reach}_{
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ignorespaces
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Lang{distance}_{
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operatorname{path}}$,}
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ignorespaces
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Lang{reach}_{
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\end_layout
\begin_layout Plain Layout
}
\end_layout
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\end_layout
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Lang{distance}$,}
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\end_layout
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}
\end_layout
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}
\end_layout
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\backslash
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\end_layout
\end_inset
\end_layout
\begin_layout BeginFrame
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
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\end_inset
\end_layout
\begin_layout Columns
\begin_inset Argument 1
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status open
\begin_layout Plain Layout
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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
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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
\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 BeginFrame
All Variants of Finding Paths in Directed Graphs
\begin_inset Newline newline
\end_inset
Are Equally Difficult
\end_layout
\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
\begin_layout AgainFrame
\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
3
\end_layout
\end_inset
hierarchy
\end_layout
\begin_layout BeginFrame
Finding Paths in Forests and Directed Paths is Easy,
\begin_inset Newline newline
\end_inset
But Not Trivial
\end_layout
\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 Separator
\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
\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 BeginFrame
Definition of the Tournament Reachability Problem
\end_layout
\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
\begin_layout BeginFrame
The Tournament Reachability Problem is Very Easy
\end_layout
\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
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\end_inset
and even
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\end_inset
.
\end_layout
\begin_layout Itemize
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\end_inset
.
\end_layout
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
2012-12-19 18:47:00 +00:00
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 BeginFrame
Finding a Shortest Path Is as Difficult as
\begin_inset Newline newline
\end_inset
the Distance Problem
\end_layout
\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
\begin_layout BeginFrame
The Tournament Distance Problem is Hard
\end_layout
\begin_layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is
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\end_inset
-complete.
\end_layout
\begin_layout Standard
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status collapsed
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\backslash
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\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
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\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
\begin_layout BeginFrame
Proof That
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\end_inset
is NL-complete
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
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\backslash
nointerlineskip
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\end_inset
\end_layout
\begin_layout Columns
\begin_inset Argument 1
2012-11-23 16:25:08 +00:00
status open
\begin_layout Plain Layout
2012-11-23 16:25:08 +00:00
t,onlytextwidth
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\end_inset
\end_layout
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status collapsed
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\backslash
setlength
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\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
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\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 Separator
\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
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\begin_layout Example
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status collapsed
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pgfnodeconnline{A3}{C2}
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pgfnodeconnline{B3}{D2}
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pgfnodeconnline{C3}{D2}
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pgfnodeconnline{D3}{A2}
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\backslash
pgfnodeconnline{D3}{B2}
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\backslash
pgfnodeconnline{A4}{C3}
\end_layout
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\backslash
pgfnodeconnline{A4}{D3}
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\backslash
pgfnodeconnline{B4}{A3}
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\backslash
pgfnodeconnline{B4}{C3}
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pgfnodeconnline{B4}{D3}
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\backslash
pgfnodeconnline{C4}{D3}
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pgfnodeconnline{D4}{B3}
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\end_layout
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pgfnodeconnline{B3}{C3}
\end_layout
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pgfnodeconnline{C3}{D3}
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pgfnodeconnline{A4}{B4}
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\backslash
pgfnodeconnline{B4}{C4}
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\end_inset
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\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
2012-12-19 18:47:00 +00:00
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 BeginFrame
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
\end_layout
\begin_layout Definition
An
\color none
\color red
approximation scheme for
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\end_inset
\color none
\color inherit
gets as input
\end_layout
\begin_deeper
\begin_layout Enumerate
a tuple
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and
\end_layout
\begin_layout Enumerate
a number
\begin_inset Formula $r>1$
\end_inset
.
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It outputs
\end_layout
\begin_layout Itemize
a path from
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\end_inset
to
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\begin_inset Formula $t$
\end_inset
of length at most
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.
\end_layout
\end_deeper
\begin_layout BeginFrame
There Exists a Logspace Approximation Scheme for
\begin_inset Newline newline
\end_inset
the Tournament Shortest Path Problem
\end_layout
\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
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\begin_inset space \hfill{}
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status collapsed
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\backslash
hyperlink{optimality}{
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status collapsed
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7
\end_layout
\end_inset
hierarchy
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If a frame includes a program listing, the frame needs to be marked as
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This is some program code:
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\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
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{frame}
\end_layout
\end_inset
\end_layout
\begin_layout Section*
Summary
\end_layout
\begin_layout Subsection*
Summary
\end_layout
\begin_layout BeginFrame
Summary
\end_layout
\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 Separator
\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
\begin_layout Subsection*
For Further Reading
\end_layout
\begin_layout BeginFrame
For Further Reading
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
beamertemplatebookbibitems
\end_layout
\end_inset
\end_layout
\begin_layout Bibliography
\labelwidthstring References
\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
\labelwidthstring References
\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
\labelwidthstring References
\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
\begin_layout EndFrame
\end_layout
\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 BeginFrame
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
[label=independence]
\end_layout
\end_inset
Definition of Independence Number of a Graph
\end_layout
\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
\begin_layout BeginFrame
The Results for Tournaments also Apply to
\begin_inset Newline newline
\end_inset
Graphs With Bounded Independence Number
\end_layout
\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 Separator
\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 Separator
\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
\begin_layout Subsection
Finding Paths in Undirected Graphs
\end_layout
\begin_layout BeginFrame
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
<1-2>[label=undirected]
\end_layout
\end_inset
The Complexity of Finding Paths in Undirected Graphs
\begin_inset Newline newline
\end_inset
Is Party Unknown.
\end_layout
\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
\begin_layout Subsection
The Approximation Scheme is Optimal
\end_layout
\begin_layout BeginFrame
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
[label=optimality]
\end_layout
\end_inset
The Approximation Scheme is Optimal
\end_layout
\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
\begin_layout Enumerate
Suppose the approximation scheme exists.
\begin_inset Newline newline
\end_inset
We show
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
\end_inset
.
\end_layout
\begin_layout Enumerate
Let
\begin_inset Formula $(T,s,t)$
\end_inset
be an input.
Let
\begin_inset Formula $n$
\end_inset
be the number of vertices.
\end_layout
\begin_layout Enumerate
Run the approximation scheme for
\begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$
\end_inset
.
\begin_inset Newline newline
\end_inset
This needs space
\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
\end_inset
.
\end_layout
\begin_layout Enumerate
The resulting path has optimal length.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
qedhere
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout EndFrame
\end_layout
\end_body
\end_document