From: Martin Mares <mj@ucw.cz>
Date: Mon, 2 Jun 2008 17:38:56 +0000 (+0200)
Subject: First bits of the abstract.
X-Git-Tag: phd-final~25
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=4784e50b5068af0705e9c8f996cd4b74622ec7de;p=saga.git

First bits of the abstract.
---

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 CHAPTERS=cover pref mst ram adv opt dyn appl rank epilog notation
 
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+\input macros.tex
+
+\def\rawchapter#1{\vensure{1in}\bigbreak\bigbreak
+\leftline{\chapfont #1}
+\bigskip
+}
+
+\chapter{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,
+both problems seem to be simple, boring and already solved, because we have poly\-nom\-ial-time
+algorithms for them since ages. But when we come closer and seek algorithms that
+are really efficient, the problems twirl and twist and withstand many a~brave
+attempt at the optimum solution. They also reveal a~vast and diverse landscape
+of a~deep and beautiful theory. Still closer, this landscape turns out to be interwoven
+with the intricate details of various models of computation and even of arithmetics
+itself.
+
+I have tried to cover all known important results on both problems and unite them
+in a~single coherent theory. At many places, I have attempted to contribute my own
+little stones to this mosaic: several new results, simplifications of existing
+ones, and last, but not least filling in important details where the original
+authors have missed some.
+
+When compared with the earlier surveys on the minimum spanning trees, most
+notably Graham and Hell \cite{graham:msthistory} and Eisner \cite{eisner:tutorial},
+this work adds many of the recent advances, the dynamic algorithms and
+also the relationship with computational models. No previous work covering
+the ranking problems in their entirety is known.
+
+The early parts of this thesis also served as a~basis for a~course on graph
+algorithms which I was teaching at our faculty during years 2006 and~2007. They are
+included in the textbook \cite{mm:ga} which I have written for this course.
+
+I~have tried to stick to the usual notation except where it was too inconvenient.
+Most symbols are defined at the place where they are used for the first time.
+To avoid piling up too many symbols at places that speak about a~single fixed graph,
+this graph is always called~$G$, its set of vertices and edges are denoted by $V$
+and~$E$ respectively, and I~also use~$n$ for the number of its vertices and $m$~for
+the number of edges. At places where there could be a~danger of confusion, more explicit notation
+is used instead.
+
+\chapter{Minimum Spanning Trees}
+
+\section{The Problem}
+
+The problem of finding a minimum spanning tree of a weighted graph is one of the
+best studied problems in the area of combinatorial optimization since its birth.
+Its colorful history (see \cite{graham:msthistory} and \cite{nesetril:history} for the full account)
+begins in~1926 with the pioneering work of Bor\o{u}vka
+\cite{boruvka:ojistem}\foot{See \cite{nesetril:boruvka} for an English translation with commentary.},
+who studied primarily an Euclidean version of the problem related to planning
+of electrical transmission lines (see \cite{boruvka:networks}), but gave an efficient
+algorithm for the general version of the problem. As it was well before the dawn of graph
+theory, the language of his paper was complicated, so we will better state the problem
+in contemporary terminology:
+
+\proclaim{Problem}Given an undirected graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$,
+find its minimum spanning tree, defined as follows:
+
+\defn\id{mstdef}%
+For a given graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$:
+\itemize\ibull
+\:A~subgraph $H\subseteq G$ is called a \df{spanning subgraph} if $V(H)=V(G)$.
+\:A~\df{spanning tree} of~$G$ is any spanning subgraph of~$G$ that is a tree.
+\:For any subgraph $H\subseteq G$ we define its \df{weight} $w(H):=\sum_{e\in E(H)} w(e)$.
+  When comparing two weights, we will use the terms \df{lighter} and \df{heavier} in the
+  obvious sense.
+\:A~\df{minimum spanning tree (MST)} of~$G$ is a spanning tree~$T$ such that its weight $w(T)$
+  is the smallest possible among all the spanning trees of~$G$.
+\:For a disconnected graph, a \df{(minimum) spanning forest (MSF)} is defined as
+  a union of (minimum) spanning trees of its connected components.
+\endlist
+
+Bor\o{u}vka's work was further extended by Jarn\'\i{}k \cite{jarnik:ojistem}, again in
+mostly geometric setting. He has discovered another efficient algorithm. However, when
+computer science and graph theory started forming in the 1950's and the
+spanning tree problem was one of the central topics of the flourishing new
+disciplines, the previous work was not well known and the algorithms had to be
+rediscovered several times.
+
+In the next 50 years, several significantly faster algorithms were discovered, ranging
+from the $\O(m\timesbeta(m,n))$ time algorithm by Fredman and Tarjan \cite{ft:fibonacci},
+over algorithms with inverse-Ackermann type complexity by Chazelle \cite{chazelle:ackermann}
+and Pettie \cite{pettie:ackermann}, to an~algorithm by Pettie \cite{pettie:optimal}
+whose time complexity is provably optimal.
+
+\chapter{Bibliography}
+
+\dumpbib
+
+\vfill\eject
+\ifodd\pageno\else\hbox{}\fi
+
+\bye