From ffb37804836177acfd7636b71724fd2fe732a295 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Wed, 4 Jun 2008 12:18:52 +0200
Subject: [PATCH] Abstract: Minor improvements.

---
 abstract.tex | 21 +++++++++++----------
 1 file changed, 11 insertions(+), 10 deletions(-)

diff --git a/abstract.tex b/abstract.tex
index ee556c8..5e1afff 100644
--- a/abstract.tex
+++ b/abstract.tex
@@ -4,20 +4,21 @@
 \advance\hsize by 1cm
 \advance\vsize by 20pt
 
-\font\chapfont=csb17
-\def\rawchapter#1{\vensure{0.5in}\bigbreak\bigbreak
+\font\chapfont=csb14 at 16pt
+\def\rawchapter#1{\vensure{0.5in}\bigskip\goodbreak
 \leftline{\chapfont #1}
 }
 
-\def\rawsection#1{\bigskip
+\def\rawsection#1{\medskip\smallskip
 \leftline{\secfont #1}
 \nobreak
 \smallskip
 \nobreak
 }
 
-\chapter{Introduction}
-\medskip
+\def\schapter#1{\chapter{#1}\medskip}
+
+\schapter{Introduction}
 
 This thesis tells the story of two well-established problems of algorithmic
 graph theory: the minimum spanning trees and ranks of permutations. At distance,
@@ -254,7 +255,7 @@ but unlike the RAM's they turn out to be equivalent up to constant slowdown.
 Our formal definition is closely related to the \em{linking automaton} proposed
 by Knuth in~\cite{knuth:fundalg}.
 
-\section{Pointer machine techniques}\id{bucketsort}%
+\section{Bucket sorting and related techniques}\id{bucketsort}%
 
 In the Contractive Bor\o{u}vka's algorithm, we needed to contract a~given
 set of edges in the current graph and then flatten the graph, all this in time $\O(m)$.
@@ -435,7 +436,7 @@ It indeed is in minor-closed classes:
 Let $\cal C$ be a graph class with density~$\varrho$ and $G\in\cal C$ a~graph
 with $n$~vertices. Then at least $n/2$ vertices of~$G$ have degree at most~$4\varrho$.
 
-So we get the following algorithm:
+This leads to the following algorithm:
 
 \algn{Local Bor\o{u}vka's Algorithm, Mare\v{s} \cite{mm:mst}}%
 \algo
@@ -605,7 +606,7 @@ finds the MSF of the original graph, but without the heavy edges.
 \algout The minimum spanning forest of~$G$.
 \endalgo
 
-A~careful analysis of this algorithm, based on properties of its recursion tree
+\>A~careful analysis of this algorithm, based on properties of its recursion tree
 and on the peak-finding algorithm of the previous section, yields the following time bounds:
 
 \thm
@@ -1406,7 +1407,7 @@ permanents in advance. When we plug it in the general algorithm, we get:
 For every~$n$, the derangements on the set~$[n]$ can be ranked and unranked according to the
 lexicographic order in time~$\O(n)$ after spending $\O(n^2)$ on initialization of auxiliary tables.
 
-\chapter{Conclusions}
+\schapter{Conclusions}
 
 We have seen the many facets of the minimum spanning tree problem. It has
 turned out that while the major question of the existence of a~linear-time
@@ -1451,7 +1452,7 @@ techniques of general interest: the unification procedures using pointer-based
 bucket sorting and the vector computations on the RAM. We hope that they will
 be useful in many other algorithms.
 
-\chapter{Bibliography}
+\schapter{Bibliography}
 
 \dumpbib
 
-- 
2.39.2