From: Martin Mares <mj@ucw.cz>
Date: Mon, 16 Jun 2008 08:35:07 +0000 (+0200)
Subject: Abstract: Minor typographic details.
X-Git-Tag: phd-final~7
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=bbc7f91c4a18336b4ae1fda3af445c6a93b1b133;p=saga.git

Abstract: Minor typographic details.
---

diff --git a/abstract.tex b/abstract.tex
index c05f657..12f2e61 100644
--- a/abstract.tex
+++ b/abstract.tex
@@ -6,7 +6,6 @@
 \advance\hsize by 1cm
 \advance\vsize by 20pt
 
-\font\chapfont=csb14 at 16pt
 \def\rawchapter#1{\vensure{0.5in}\bigskip\goodbreak
 \leftline{\chapfont #1}
 }
@@ -210,8 +209,8 @@ and also to analyse:
 \endalgo
 
 \thm
-The Contractive Bor\o{u}vka's algorithm finds the MST of the input graph in
-time $\O(\min(n^2,m\log n))$.
+The Contractive Bor\o{u}vka's algorithm finds the MST of the graph given as
+its input in time $\O(\min(n^2,m\log n))$.
 
 We also show that this time bound is tight --- we construct an~explicit
 family of graphs on which the algorithm spends $\Theta(m\log n)$ steps.
@@ -671,6 +670,7 @@ of non-overlapping contractible subgraphs called \df{clusters} and we put aside
 We recursively compute the MSF of those subgraphs and of the contracted graph. Then we take the
 union of these MSF's and add the corrupted edges. According to the previous lemma, this does not produce
 the MSF of~$G$, but a~sparser graph containing it, on which we can continue.
+%%The following theorem describes the properties of this partition:
 
 \thmn{Partitioning to contractible clusters, Chazelle \cite{chazelle:almostacker}}\id{partthm}%
 Given a~weighted graph~$G$ and parameters $\varepsilon$ ($0<\varepsilon\le 1/2$)
@@ -691,7 +691,8 @@ and~$t$, we can construct a~collection $\C=\{C_1,\ldots,C_k\}$ of clusters and a
 
 The Pettie's and Ramachandran's algorithm combines the idea of robust partitioning with optimal decision
 trees constructed by brute force for very small subgraphs.
-Formally, the decision trees are defined as follows:
+%%Formally, the decision trees are defined as follows:
+Let us define them first:
 
 \defnn{Decision trees and their complexity}\id{decdef}%
 A~\df{MSF decision tree} for a~graph~$G$ is a~binary tree. Its internal vertices
diff --git a/fonts10.tex b/fonts10.tex
index 9aff272..7bf40d7 100644
--- a/fonts10.tex
+++ b/fonts10.tex
@@ -61,7 +61,7 @@
 \rm
 
 % Other fonts
-\font\chapfont=csssdc17 scaled \magstep1
+\font\chapfont=csb14 at 16pt
 \font\secfont=csb14
 \font\secitfont=csbxti14