+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 and it can be said
+that it stood at the cradle of this discipline. Its colorful history (see \cite{graham:msthistory}
+and \cite{nesetril:history} for the full account) begins in~1926 with
+the pioneering work of Bor\accent23uvka
+\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 birth of graph
+theory, the language of his paper was complicated, so we will rather 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, where:
+
+\defn For a given graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$:
+\itemize\ibull
+\:A~tree $T$ is a \df{spanning tree} of~$G$ if and only if $V(T)=V(G)$ and $E(T)\subseteq E(G)$.
+\:For any subgraph $H\subseteq G$ we define its \df{weight} $w(H):=\sum_{e\in E(H)} w(e)$.
+\:A~spanning tree~$T$ is \df{minimal} iff $w(T)$ is the smallest possible of all spanning trees.
+\endlist
+
+Bor\accent23uvka's work was further extended by Jarn\'\i{}k \cite{jarnik:ojistem}, again in
+mostly geometric setting, giving another polynomial 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 have been
+rediscovered several times.
+
+Recently, several significantly faster algorithms were discovered, most notably the
+$\O(m\beta(m,n))$-time algorithm by Fredman and Tarjan \cite{ft:fibonacci} and
+algorithms with inverse-Ackermann type complexity by Chazelle \cite{chazelle:ackermann}
+and Pettie \cite{pettie:ackermann}.
+
+\FIXME{Write the rest of the history.}
+
+This chapter attempts to survery the important algorithms for finding the MST and it
+also presents several new ones.
+
+\section{Basic Properties}
+
+In this section, we will examine the basic properties of spanning trees and prove
+several important theorems to base the algorithms upon. We will follow the theory
+developed by Tarjan in~\cite{tarjan:dsna}.
+
+First of all, let us show that the weights on edges are not necessary for the
+definition of the MST. We can formulate an equivalent characterization using
+an ordering of edges instead.
+
+\defn For a graph~$G$ and its tree subgraph~$T$, we define:
+\itemize\ibull
+\:For vertices $x$ and $y$, let $T[x,y]$ denote the (unique) path in~$T$ joining $x$ and~$y$.
+\:For an edge $e=xy$ outside~$T$, we will call $T[e]:=T[x,y]$ the \df{path covered by~$e$} and
+ the edges of this path \df{edges covered by~$e$}.
+\:An edge~$e=xy$ outside~$T$ is called \df{$T$-light} if it covers a heavier edge, i.e., if there
+ is an edge $f\in T[e]$ such that $w(f) > w(e)$.
+\:An edge~$e$ outside~$T$ is called \df{$T$-heavy} if it is not light.
+\endlist
+
+\theorem A~spanning tree~$T$ of a graph~$G$ is minimal iff there is no $T$-light edge in~$G$.
+
+To prove this theorem, we will use a notion of edge exchanges, similar to Steinitz theorem
+in linear algebra or the exchanges in matroid theory.
+\FIXME{reference}
+
+\defn For a tree~$T$ and edges $e\in T$ and $f\not\in T$, the \df{exchange}
+XXXXXX
+
+For the other implication, we will make use of the following lemmata, again
+based on exchange of edges:
+
+{\narrower
+\lemman{Exchange property of spanning trees}
+Let $T$ and $T'$ be spanning trees of a common graph~$G$. Then there exists
+a sequence
+
+}
+
+Back to the proof of the theorem:
+
+The implication $\Rightarrow$ is straightforward: If there is a $T$-light edge~$e$, there
+exists an edge $f\in T[e]$ such that $w(f)>w(e)$. Now $T-f$ is a forest of two trees
+with endpoints of~$e$ located in different components, so adding $e$ to this forest
+must restore connectivity and $T'=T-f+e$ is another spanning tree with weight $w(T') = w(T)-w(f)+w(e) < w(T)$.
+Hence $T$ could not have been minimal.
+
+
+%\:For a disconnected graph~$G$, we define the \df{(minimal) spanning forest (or MSF)}
+% as an union of (minimal) spanning trees of its connected components.
projects / saga.git / commitdiff