\proclaim{Problem}Given an undirected graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$,
find its minimum spanning tree, defined as follows:
-\defn\thmid{mstdef}%
+\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 $V(H)=V(G)$.
definition of the MST. We can formulate an equivalent characterization using
an ordering of edges instead.
-\defnn{Heavy and light edges}\thmid{heavy}%
+\defnn{Heavy and light edges}\id{heavy}%
Let~$T$ be a~spanning tree. Then:
\itemize\ibull
\:For vertices $x$ and $y$, let $T[x,y]$ denote the (unique) path in~$T$ joining $x$ and~$y$.
Please note that the above properties also apply to tree edges
which by definition cover only themselves and therefore they are always heavy.
-\lemman{Light edges}\thmid{lightlemma}%
+\lemman{Light edges}\id{lightlemma}%
Let $T$ be a spanning tree. If there exists a $T$-light edge, then~$T$
is not minimum.
= w(T)-w(e')+w(e) < w(T)$. Hence $T$ could not have been minimum.
\qed
-\figure{mst2.eps}{278pt}{An edge exchange as in the proof of Lemma~\thmref{lightlemma}}
+\figure{mst2.eps}{278pt}{An edge exchange as in the proof of Lemma~\ref{lightlemma}}
The converse of this lemma is also true and to prove it, we will once again use
technique of transforming trees by \df{exchanges} of edges. In the proof of the
a spanning tree. In fact, it is possible to transform any spanning tree
to any other spanning tree by a sequence of exchanges.
-\lemman{Exchange property for trees}\thmid{xchglemma}%
+\lemman{Exchange property for trees}\id{xchglemma}%
Let $T$ and $T'$ be spanning trees of a common graph. Then there exists
a sequence of edge exchanges which transforms $T$ to~$T'$. More formally,
there exists a sequence of spanning trees $T=T_0,T_1,\ldots,T_k=T'$ such that
hypothesis to $T^*$ and $T'$ to get the rest of the exchange sequence.
\qed
-\figure{mst1.eps}{295pt}{One step of the proof of Lemma~\thmref{xchglemma}}
+\figure{mst1.eps}{295pt}{One step of the proof of Lemma~\ref{xchglemma}}
-\lemman{Monotone exchanges}\thmid{monoxchg}%
+\lemman{Monotone exchanges}\id{monoxchg}%
Let $T$ be a spanning tree such that there are no $T$-light edges and $T'$
be an arbitrary spanning tree. Then there exists a sequence of edge exchanges
transforming $T$ to~$T'$ such that the weight does not increase in any step.
than~$e'$ as $e'$ was not $T$-light.
\qed
-\thm\thmid{mstthm}%
+\thm\id{mstthm}%
A~spanning tree~$T$ is minimum iff there is no $T$-light edge.
\proof
-If~$T$ is minimum, then by Lemma~\thmref{lightlemma} there are no $T$-light
+If~$T$ is minimum, then by Lemma~\ref{lightlemma} there are no $T$-light
edges.
Conversely, when $T$ is a spanning tree without $T$-light edges
and $T_{min}$ is an arbitrary minimum spanning tree, then according to the Monotone
-exchange lemma (\thmref{monoxchg}) there exists a non-decreasing sequence
+exchange lemma (\ref{monoxchg}) there exists a non-decreasing sequence
of exchanges transforming $T$ to $T_{min}$, so $w(T)\le w(T_{min})$
and thus $T$~is also minimum.
\qed
\proof
Consider two minimum spanning trees $T_1$ and~$T_2$. According to the previous
theorem, there are no light edges with respect to neither of them, so by the
-Monotone exchange lemma (\thmref{monoxchg}) there exists a sequence of non-decreasing
+Monotone exchange lemma (\ref{monoxchg}) there exists a sequence of non-decreasing
edge exchanges going from $T_1$ to $T_2$. As all edge weights all distinct,
these edge exchanges must be in fact strictly increasing. On the other hand,
we know that $w(T_1)=w(T_2)$, so the exchange sequence must be empty and indeed
$T_1$ and $T_2$ must be identical.
\qed
-\rem\thmid{edgeoracle}%
+\rem\id{edgeoracle}%
To simplify the description of MST algorithms, we will expect that the weights
of all edges are distinct and that instead of numeric weights (usually accompanied
by problems with representation of real numbers in algorithms) we will be given
original graph. In the few cases where we need a more concrete input, we will
explicitly state so.
-\nota\thmid{mstnota}%
+\nota\id{mstnota}%
When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
its unique minimum spanning tree.
\algn{Red-Blue Meta-Algorithm}
\algo
-\algin A~graph $G$ with an edge comparison oracle (see \thmref{edgeoracle})
+\algin A~graph $G$ with an edge comparison oracle (see \ref{edgeoracle})
\:In the beginning, all edges are colored black.
\:Apply rules as long as possible:
\::Either pick a cut~$C$ such that its lightest edge is not blue \hfil\break and color this edge blue, \cmt{Blue rule}
reachable from~$x$ using only blue edges. If $y$~lies in~$M$, then $e$ together
with some blue path between $x$ and $y$ forms a cycle and it must be the heaviest
edge on this cycle. This holds because all blue edges have been already proven
-to be in $T_{min}$ and there can be no $T_{min}$-light edges (see Theorem~\thmref{mstthm}).
+to be in $T_{min}$ and there can be no $T_{min}$-light edges (see Theorem~\ref{mstthm}).
In this case we can apply the red rule.
On the other hand, if $y\not\in M$, then the cut formed by all edges between $M$
which is a contradiction. (Note that distinctness of edge weights was crucial here.)
\qed
-\lemma\thmid{boruvkadrop}%
+\lemma\id{boruvkadrop}%
In each iteration of the algorithm, the number of trees in~$T$ drops at least twice.
\proof
\cor
The algorithm stops in $\O(\log n)$ iterations.
-\lemma\thmid{boruvkaiter}%
+\lemma\id{boruvkaiter}%
Each iteration can be carried out in time $\O(m)$.
\proof
Jarn\'\i{}k's algorithm finds the MST of the graph in time $\O(m\log n)$.
\rem
-We will show several faster implementations in section \secref{fibonacci}.
+We will show several faster implementations in section \ref{fibonacci}.
\algn{Kruskal \cite{kruskal:mst}, the Greedy algorithm}
\algo
triangle is contracted. Either we unify the other two edges to a single edge
or we keep them as two parallel edges, leaving us with a~multigraph. We will
use the multigraph version and show that we can easily reduce the multigraph
-to a simple graph later. (See \thmref{contract} for the exact definitions.)
+to a simple graph later. (See \ref{contract} for the exact definitions.)
We only need to be able to map edges of the contracted graph to the original
edges, so each edge will carry a unique label $\ell(e)$ which will be preserved by
contractions.
-\lemman{Flattening a multigraph}\thmid{flattening}%
+\lemman{Flattening a multigraph}\id{flattening}%
Let $G$ be a multigraph and $G'$ its subgraph such that all loops have been
removed and each bundle of parallel edges replaced by its lightest edge.
Then $G'$~has the same MST as~$G$.
\proof
The only non-trivial parts are steps 6 and~7. Contractions can be handled similarly
-to the unions in the original Bor\o{u}vka's algorithm (see \thmref{boruvkaiter}).
+to the unions in the original Bor\o{u}vka's algorithm (see \ref{boruvkaiter}).
We build an auxillary graph containing only the selected edges~$e_i$, find
connected components of this graph and renumber vertices in each component to
the identifier of the component. This takes $\O(m)$ time.
$m_i$'s are decreasing exponentially.
The number of trees in the non-contracting version of the algorithm decreases
-at least twice in each iteration (Lemma \thmref{boruvkadrop}) and the
+at least twice in each iteration (Lemma \ref{boruvkadrop}) and the
same must hold for the number of vertices in the contracting version.
Therefore $n_i\le n/2^i$.
\proof
% We seem not to need this lemma for multigraphs...
%If there are any loops or parallel edges in~$G$, we can flatten the graph. According to the
-%Flattening lemma (\thmref{flattening}), the MST stays the same and if we remove a parallel edge
+%Flattening lemma (\ref{flattening}), the MST stays the same and if we remove a parallel edge
%or loop~$f$, then $\pi(f)$ would be removed when flattening~$G/e$, so $f$ never participates
%in a MST.
The right-hand side of the equality is a spanning tree of~$G$, let us denote it by~$T$ and
the MST of $G/e$ by~$T'$. If $T$ were not minimum, there would exist a $T$-light edge~$f$ in~$G$
-(according to Theorem \thmref{mstthm}). If the path $T[f]$ covered by~$f$ does not contain~$e$,
+(according to Theorem \ref{mstthm}). If the path $T[f]$ covered by~$f$ does not contain~$e$,
then $\pi[T[f]]$ is a path covered by~$\pi(f)$ in~$T'$. Otherwise $\pi(T[f]-e)$ is such a path.
In both cases, $f$ is $T'$-light, which contradicts the minimality of~$T'$. (We do not have
a~multigraph version of the theorem, but this direction is a straightforward edge exchange,
\defn
A~graph~$H$ is a \df{minor} of a~graph~$G$ iff it can be obtained
-from a subgraph of~$G$ by a sequence of graph contractions (see \thmref{simpcont}).
+from a subgraph of~$G$ by a sequence of graph contractions (see \ref{simpcont}).
\defn
A~class~$\cal C$ of graphs is \df{minor-closed}, when for every $G\in\cal C$ and
embeddable in any fixed surface. Many nice properties of planar graphs extend
to these classes, too, most notable the linearity of the number of edges.
-\defn\thmid{density}%
+\defn\id{density}%
Let $\cal C$ be a class of graphs. We define its \df{edge density} $\varrho(\cal C)$
to be the infimum of all~$\varrho$'s such that $\vert E(G) \vert \le \varrho\cdot\vert V(G)\vert$
holds for every $G\in\cal C$.
\section{Using Fibonacci heaps}
-\secid{fibonacci}
+\id{fibonacci}
% G has to be connected, so m=O(n)
% mention Steiner trees
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