Notation:
-* impedance mismatch in terminology: contraction of G along e vs. contraction of e.
- use \delta(X) notation
- unify use of n(G) vs. n
-* change the notation for contractions -- use double slash instead of the dot?
- introduce \widehat\O early
Typography:
\:While $n(G)>1$:
\::While there exists a~vertex~$v$ such that $\deg(v)\le t$:
\:::Select the lightest edge~$e$ incident with~$v$.
-\:::Contract~$G$ along~$e$.
+\:::Contract~$e$ in~$G$.
\:::$T\=T + \ell(e)$.
\::Flatten $G$, removing parallel edges and loops.
\algout Minimum spanning tree~$T$.
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\o{u}vka
+them to increase the density of the graph. 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}
We start with an~arbitrary vertex, grow the tree as usually and once the heap gets too large,
we conserve the current tree and start with a~different vertex and an~empty heap. When this
process runs out of vertices, it has identified a~sub-forest of the MST, so we can
-contract the graph along the edges of~this forest and iterate.
+contract the edges of~this forest and iterate.
\algn{Iterated Jarn\'\i{}k; Fredman and Tarjan \cite{ft:fibonacci}}\id{itjar}%
\algo
\:::Denote the resulting tree~$R$.
\:::$F\=F\cup R$.
\::$T\=T\cup \ell[F]$. \cmt{Remember MST edges found in this phase.}
-\::Contract~$G$ along all edges of~$F$ and flatten it.
+\::Contract all edges of~$F$ and flatten~$G$.
\algout Minimum spanning tree~$T$.
\endalgo
\def\E{{\bb E}}
\def\crpt{\mathbin{\Uparrow}}
\def\C{{\cal C}}
+\def\sgc{\mathbin{.}}
\def\brk{\hfil\break}
While the classical algorithms are based on growing suitable trees, they
can be also reformulated in terms of edge contraction. Instead of keeping
-a forest of trees, we can keep each tree contracted to a single vertex.
+a~forest of trees, we can keep each tree contracted to a single vertex.
This replaces the relatively complex tree-edge incidencies by simple
vertex-edge incidencies, potentially speeding up the calculation at the
expense of having to perform the contractions.
\:While $n(G)>1$:
\::For each vertex $v_k$ of~$G$, let $e_k$ be the lightest edge incident to~$v_k$.
\::$T\=T\cup \{ \ell(e_k) \}$. \cmt{Remember labels of all selected edges.}
-\::Contract $G$ along all edges $e_k$, inheriting labels and weights.\foot{In other words, we ask the comparison oracle for the edge $\ell(e)$ instead of~$e$.}
+\::Contract all edges $e_k$, inheriting labels and weights.\foot{In other words, we ask the comparison oracle for the edge $\ell(e)$ instead of~$e$.}
\::Flatten $G$, removing parallel edges and loops.
\algout Minimum spanning tree~$T$.
\endalgo
\lemman{Contraction of MST edges}\id{contlemma}%
Let $G$ be a weighted graph, $e$~an arbitrary edge of~$\mst(G)$, $G/e$ the multigraph
-produced by contracting $G$ along~$e$, and $\pi$ the bijection between edges of~$G-e$ and
+produced by contracting~$e$ in~$G$, and $\pi$ the bijection between edges of~$G-e$ and
their counterparts in~$G/e$. Then: $$\mst(G) = \pi^{-1}[\mst(G/e)] + e.$$
\proof
\n{$G[U]$}{subgraph induced by a~set $U\subset V(G)$}
\n{$X \choose k$}{the set of all $k$-element subsets of a set~$X$}
\n{$G/e$}{multigraph contraction \[contract]}
-\n{$G.e$}{simple graph contraction \[simpcont]}
+\n{$G\sgc e$}{simple graph contraction \[simpcont]}
\n{$G/X$, $G.X$}{contraction by a~set $X$ of vertices or edges \[setcont]}
\n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) \mid x\in X \}$}
\n{$f[e]$}{as edges are two-element sets, $f[e]$ maps both endpoints of an edge~$e$}
the same objects, although they formally differ.
\defn\id{contract}%
-Let $G=(V,E,M)$ be a multigraph and $e=xy$ its edge. \df{(Multigraph) contraction of~$G$ along~$e$}
+Let $G=(V,E,M)$ be a multigraph and $e=xy$ its arbitrary edge.
+The \df{(multigraph) contraction of~$e$ in~$G$}
produces a multigraph $G/e=(V',E',M')$ such that:
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
}$$
-Sometimes we need contraction for simple graphs as well. It corresponds to performing
-the multigraph contraction, unifying parallel edges and deleting loops.
+Sometimes we need contraction for simple graphs as well. It is equivalent to performing
+the multigraph contraction and then unifying parallel edges and deleting loops.
\defn\id{simpcont}%
-Let $G=(V,E)$ a simple graph and $e=xy$ its edge. \df{(Simple graph) contraction of~$G$ along~$e$}
-produces a graph $G.e=(V',E')$ such that:
+Let $G=(V,E)$ a simple graph and $e=xy$ its arbitrary edge.
+The \df{(simple graph) contraction of~$e$ in~$G$}
+produces a graph $G\sgc e=(V',E')$ such that:
$$\eqalign{
V' &= (V(G) \setminus \{x,y\}) \cup \{v_e\},\quad\hbox{where $v_e$ is a new vertex,}\cr
E' &= \{ \{m(x),m(y)\} \mid xy\in E \land m(x)\ne m(y) \},\cr
-m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr v & \hbox{otherwise.}} \cr
+m(x) &= \cases{v_e & \hbox{for $v=x,y,$}\cr \noalign{\vskip5pt} v & \hbox{otherwise.}} \cr
}$$
\defn\id{setcont}%
-We can also extend the above definitions to contractions by a~set of vertices or edges.
+We can also extend the above definitions to contractions of a~set of vertices or edges.
For $F\subseteq E(G)$, the graph $G/F$ is defined as $(G/f_1)/f_2/\ldots/f_k$ where
$f_1,\ldots,f_k$ are the elements of~$F$ (you can observe that the result
does not depend on the order of edges). For $U\subseteq V(G)$, we define $G/U$
as the graph with all vertices of~$U$ merged to a~single vertex, that is $(G\cup U^*)/U^*$,
-where $U^*$ is the complete graph on~$U$. Similarly for $G.F$ and $G.U$.
+where $U^*$ is the complete graph on~$U$. Similarly for $G\sgc F$ and $G\sgc U$.
%--------------------------------------------------------------------------------
projects / saga.git / commitdiff