- move TOC to the beginning of the book
- mention notation
+- cite GA booklet
Spanning trees:
- cite Eisner's tutorial \cite{eisner:tutorial}
-- move the remark on disconnected graphs? separate section?
-- mention graphs with non-unique weights? also in the separate section?
- Some algorithms (most notably Fredman-Tarjan) do not need flattening
-- citation of mixed Boruvka-Jarnik
- use the notation for contraction by a set
- mention bugs in Valeria's verification paper
- more references on decision trees
-- introduce Cut rule and Cycle rule earlier
- Lemma: deletion of a non-MST edge does not alter the MST
+- mention that there are only a few algorithms based on the Red rule
Related:
- practical considerations: katriel:cycle, moret:practice (mention pairing heaps)
Global:
-- Intro: cite GA booklet
- each chapter should make clear in which model we work
- clean up bibliography
algorithm terminates after $\O(\log n)$ iterations.
Each selected edge belongs to $\mst(G)$, because it is the lightest edge of
-the trivial cut $\delta(v)$ (see the Blue Rule in \ref{rbma}).
+the trivial cut $\delta(v)$ (see the Blue rule, Lemma \ref{rbma}).
The steps 6 and~7 therefore correspond to the operation
-described by the Lemma on contraction of MST edges (\ref{contlemma}) and when
+described by the Contraction Lemma (\ref{contlemma}) and when
the algorithm stops, $T$~is indeed the minimum spanning tree.
It remains to analyse the time complexity of the algorithm. Since $G_i\in{\cal C}$, we have
As we already noted, the improved Jarn\'\i{}k's algorithm runs in linear time
for sufficiently dense graphs. In some cases, it is useful to combine it with
another MST algorithm, which identifies a~part of the MST edges and contracts
-the graph to increase its density. For example, we can perform several Bor\v{u}vka
+the graph to increase its density. For example, we can perform several Bor\o{u}vka
steps and find the rest of the MST by the Active Edge Jarn\'\i{}k's algorithm.
\algn{Mixed Bor\o{u}vka-Jarn\'\i{}k}
Selecting a~subgraph at random will unavoidably produce disconnected subgraphs
at occassion, so we will drop the implicit assumption that all graphs are
connected for this section and we will always search for the minimum spanning forest.
-As we already noted (Remark \ref{disconn}), with a~little bit of care our
+As we already noted (\ref{disconn}), with a~little bit of care our
algorithms and theorems keep working.
Since we need the MST verification algorithm for finding the $T$-heavy edges,
\cor
The algorithm stops in $\O(\log n)$ iterations.
-\lemma
+\lemma\id{borcorr}%
Bor\o{u}vka's algorithm outputs the MST of the input graph.
\proof
Every spanning tree of~$G'$ is a spanning tree of~$G$. In the other direction:
Loops can be never contained in a spanning tree. If there is a spanning tree~$T$
containing a~removed edge~$e$ parallel to an edge~$e'\in G'$, exchaning $e'$
-for~$e$ makes~$T$ lighter. \qed
-
-\rem Removal of the heavier of a pair of parallel edges can be also viewed
-as an application of the Red rule on a two-edge cycle. And indeed it is, the
-Red-Blue procedure works on multigraphs as well as on simple graphs and all the
-classical algorithms also do. We would only have to be more careful in the
-formulations and proofs, which we preferred to avoid.
+for~$e$ makes~$T$ lighter. (This is indeed the multigraph version of the Red
+lemma applied to a~two-edge cycle, as we will see in \ref{multimst}.)
+\qed
\algn{Contractive version of Bor\o{u}vka's algorithm}\id{contbor}
\algo
Finally, we will show a family of graphs where the $\O(m\log n)$ bound on time complexity
is tight. The graphs do not have unique weights, but they are constructed in a way that
the algorithm never compares two edges with the same weight. Therefore, when two such
-graphs are monotonely isomorphic (see~\ref{mstiso}), the algorithm processes them in the same way.
+graphs are monotonically isomorphic (see~\ref{mstiso}), the algorithm processes them in the same way.
\defn
A~\df{distractor of order~$k$,} denoted by~$D_k$, is a path on $n=2^k$~vertices $v_1,\ldots,v_n$
edges as every $H_{a,k}$ contains a complete graph on~$a$ vertices.
\qed
-\remn{Disconnected graphs}\id{disconn}%
+%--------------------------------------------------------------------------------
+
+\section{Lifting restrictions}
+
+In order to have a~simple and neat theory, we have introduced several restrictions
+on the graphs in which we search for the MST. As in some rare cases we are going to
+meet graphs that do not fit into this simplified world, let us quickly examine what
+happens when the restrictions are lifted.
+
+\paran{Disconnected graphs}\id{disconn}%
The basic properties of minimum spanning trees and the algorithms presented in
this chapter apply to minimum spanning forests of disconnected graphs, too.
The proofs of our theorems and the steps of our algorithms are based on adjacency
$F$-light. Again, a~spanning forest~$F$ is minimum iff there are no $F$-light
edges.
+\paran{Multigraphs}\id{multimst}%
+All theorems and algorithms from this chapter work for multigraphs as well,
+only the notation sometimes gets crabbed, which we preferred to avoid. The Minimality
+theorem and the Blue rule stay unchanged. The Red rule is naturally extended to
+self-loops (which are never in the MST) and two-edge cycles (where the heavier
+edge can be dropped) as already suggested in the Flattening lemma (\ref{flattening}).
+
+\paran{Multiple edges of the same weight}\id{multiweight}%
+In case when the edge weights are not distinct, the characterization of minimum
+spanning trees using light edges is still correct, but the MST is no longer unique
+(as already mentioned, there can be as much as~$n^{n-2}$ MST's).
+
+In the Red-Blue procedure, we have to avoid being too zealous. The Blue lemma cannot
+guarantee that when a~cut contains multiple edges of the minimum weight, all of them
+are in the MST. It will however tell that if we pick one of these edges, an~arbitrary
+MST can be modified to another MST that contains this edge. Therefore the Blue rule
+will change to ``Pick a~cut~$C$ such that it does not contain any blue edge and color
+one of its lightest edges blue.'' The Red lemma and the Red rule can be handled
+in a~similar manner. The modified algorithm will be then guaranteed to find one of
+the possible MST's.
+
+The Kruskal's and Jarn\'\i{}k's algorithms keep working. This is however not the case of the
+Bor\o{u}vka's algorithm, whose proof of correctness in Lemma \ref{borcorr} explicitly referred to
+distinct weights and indeed, if they are not distinct, the algorithm will occasionally produce
+cycles. To avoid the cycles, the ties in edge weight comparisons have to be broken in a~systematic
+way. The same applies to the contractive version of this algorithm.
+
\endpart
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