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1470a7b864
lyx_mirror/lib/examples/Presentations/Beamer_%28Complex%29.lyx
Scott Kostyshak a15303b938 Update format of docs using LyX binary 
Especially after the change to use semantic linefeeds (7b23c76b),
the diffs are large and it's hard to figure out what diff is the
result of the linefeed change and which diff is the result of an
edit.

By updating the docs, it will make the edits easier to understand
from the diff.

This commit used the LyX binary to write the new .lyx files since
lyx2lyx does not apply semantic linefeeds.

I used the following command:

  ./development/tools/updatedocs.py [path to 'lyx' binary]

A few documents were not updated (e.g., the command sequence used
failed because a dialog about a missing dependency was shown).

I did not update Additional.lyx for any language since it is
undergoing changes. I also didn't change anything in doc/fr since JP
is working on some of those. I also didn't change doc/ru since it
appears Yuriy already updated those.
2022-12-11 11:16:28 -05:00

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#LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 612
\begin_document
\begin_header
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The Complexity of
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Finding Paths in Tournaments
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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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Outline
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\begin_layout Section
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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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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Tournaments Arise Naturally in Different Situations
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Applications in Ordering Theory
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Applications in Sociology
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Several candidates apply for a position.
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Reviewers decide for any two candidates whom they prefer.
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Applications in Structural Complexity Theory
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A language
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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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the Optimization Problem
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the Approximation Problem
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2,4,6,8,10
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Variants of Path Finding Problems
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2-
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Reachability
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Approximation Problem:
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4-
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Construction
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\begin_inset Formula $t$
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6-
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Optimization
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\begin_inset Formula $t$
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8-
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Distance
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\begin_inset space ~
\end_inset
\begin_inset Formula $t$
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\begin_inset Formula $d$
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10-
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Approximation
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\end_inset
\begin_inset Formula $t$
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of length
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\end_deeper
\end_deeper
\end_deeper
\begin_layout Section
Review
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\begin_layout Subsection
Standard Complexity Classes
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The Classes L and NL are Defined via
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Logspace Turing Machines
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}
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\begin_layout Frame
\begin_inset Argument 4
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Logspace Turing Machines Are Quite Powerful
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\end_inset
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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
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\end_layout
\begin_layout Block
\begin_inset Argument 2
status collapsed
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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
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\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 1
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1
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\end_inset
\begin_inset Argument 3
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\begin_layout Plain Layout
label=hierarchy
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\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
The Complexity Class Hierarchy
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\end_inset
\end_layout
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\begin_inset ERT
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\end_layout
\begin_layout Plain Layout
\backslash
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\begin_layout Plain Layout
\backslash
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\begin_layout Plain Layout
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}
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\begin_layout Plain Layout
\backslash
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\backslash
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}
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\begin_layout Plain Layout
\backslash
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\begin_layout Plain Layout
\backslash
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\backslash
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\backslash
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\backslash
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\backslash
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\begin_layout Plain Layout
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\begin_layout Plain Layout
\backslash
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\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
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\end_layout
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\end_layout
\begin_layout Plain Layout
\backslash
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\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
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\begin_layout Plain Layout
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\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
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\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
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\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
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\end_layout
\begin_layout Plain Layout
\backslash
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\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
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\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}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A2}{B2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{C2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C2}{D2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A3}{B3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{C3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C3}{D3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{A4}{B4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B4}{C4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{C4}{D4}
\end_layout
\begin_layout Plain Layout
\backslash
pgfclearstartarrow
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{A3}{A1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{A4}{A2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{A4}{A1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{B3}{B1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{B4}{B2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{B4}{B1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{C3}{C1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{C4}{C2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{C4}{C1}{135}{-135}{15pt}{15pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{D3}{D1}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{D4}{D2}{135}{-135}{10pt}{10pt}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconncurve{D4}{D1}{135}{-135}{15pt}{15pt}
\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
}
\end_layout
\begin_layout Plain Layout
\backslash
only<3->{%
\end_layout
\begin_layout Plain Layout
\backslash
color<3>{red}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B1}{A2}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B2}{A3}
\end_layout
\begin_layout Plain Layout
\backslash
pgfnodeconnline{B3}{A4}
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
\backslash
only<4->{%
\end_layout
\begin_layout Plain Layout
\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"
literal "true"
\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"
literal "true"
\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"
literal "true"
\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
\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
\end_deeper
\end_body
\end_document
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