special cases. At several places, we will try to contribute our little stones to this
mosaic.
+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.
+
+\nota
+We 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.
+A~complete index of symbols with pointers to their definitions is then available
+in Appendix~\ref{notapp}. This appendix also describes the formalism of
+multigraphs and of the Ackermann's function, both of which are not defined
+consistently in the common literature.
+
+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.
+
%--------------------------------------------------------------------------------
\section{Basic properties}\id{mstbasics}%
+++ /dev/null
-\ifx\endpart\undefined
-\input macros.tex
-\fi
-
-\unchapter{Preface}
-
-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.
-
-\def\ss#1{\medskip\>{\bo #1}\enspace\eatspaces}
-
-\ss{My original results}
-
-\itemize\ibull
-\:The lower bound in Section \ref{contalg}. Not published yet.
-\:The tree isomorphism algorithm in Section \ref{bucketsort}. Not published yet.
-\:One of the algorithms for minor-closed graph classes in Section \ref{minorclosed}. Published in \cite{mm:mst}.
-\:The linear-time verification algorithm in Section \ref{verifysect} is a~simplification
- of the algorithm of King \cite{king:verifytwo} and it corrects many omissions
- in the original paper. Not published yet.
-\:The ranking algorithms in Sections \ref{ranksect} to \ref{kpranksect} are results of joint research with Milan Straka.
- Published in \cite{mm:rank}.
-\:The remaining sections of Chapter \ref{rankchap} contain unpublished original results.
-\endlist
-
-\ss{Other minor contributions}
-
-\itemize\ibull
-\:The flattening procedure in Section \ref{bucketsort}. Included in \cite{mm:mst}.
-\:The unified view of vector computations in Section \ref{bitsect}. Published
- in the textbook \cite{mm:ga}. The main ideas of this section were also included
- in the yearbook of the Czech Mathematical Olympiad \cite{horak:mofivefour}.
-\:Slight simplifications of the soft heaps and their analysis in Section \ref{shsect}.
-\:The dynamic MST algorithm for graphs with limited edge weights in Section \ref{dynmstsect}.
-\endlist
-
-\vfill\eject
-
-\ss{Acknowledgements}
-
-First of all, I~would like to thank my supervisor, Jaroslav Ne\v{s}et\v{r}il, for
-introducing me to the world of discrete mathematics and gently guiding my attempts
-to explore it with his deep insight. I~am very grateful to all members of the
-Department of Applied Mathematics and the Institute for Theoretical Computer
-Science for the work environment which was friendly and highly inspiring.
-I~cannot forget the participants of the department's seminars,
-who have listened to my talks and provided lots of important feedback.
-I~also send my acknowledgements to the members of the Math department at ETH Z\"urich and of DIMACS
-at the Rutgers University (especially to J\'anos Koml\'os) where I~spent several
-pleasant months working on what finally become a~part of this thesis.
-
-I~also thank to my family for supporting me during the plentiful years of my study,
-to my girlfriend Ani\v{c}ka for lots of patience when I~was caught up by my work and
-hardly speaking at all, to all the polar bears of Kobylisy for their furry presence, and
-finally to our cats Minuta and Dami\'an for their mastership in hiding my
-papers, which has frequently forced me to think of new ways of looking at problems
-when the old ones were impossible to find.
-
-\ss{Notation}
-
-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.
-A~complete index of symbols with pointers to their definitions is then available
-in Appendix~\ref{notapp}. This appendix also describes the formalism of
-multigraphs and of the Ackermann's function, both of which are not defined
-consistently in the common literature.
-
-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.
-
-
-\bigskip
-
-So, my gentle reader, let us nestle deep in an~ancient wing armchair. The saga of the
-graph algorithms begins~\dots
-
-\endpart
projects / saga.git / commitdiff