The contractive algorithm given in Section~\ref{contalg} has been found to perform
well on planar graphs, but in the general case its time complexity was not linear.
-Can we find any broader class of graphs where this algorithm is still efficient?
+Can we find any broader class of graphs where this algorithm is still linear?
The right context turns out to be the minor-closed graph classes, which are
closed under contractions and have bounded density.
\thmn{Excluded minors, Robertson \& Seymour \cite{rs:wagner}}
For every non-trivial minor-closed graph class~$\cal C$ there exists
a~finite set~$\cal H$ of graphs such that ${\cal C}=\Forb({\cal H})$.
+\qed
-\proof
This theorem has been proven in a~long series of papers on graph minors
culminating with~\cite{rs:wagner}. See this paper and follow the references
to the previous articles in the series.
-\qed
\para
For analysis of the contractive algorithm,
Let us return to the analysis of our algorithm.
-\thmn{MST on minor-closed classes, Mare\v{s} \cite{mm:mst}}\id{mstmcc}%
+\thmn{MST on minor-closed classes, Tarjan \cite{tarjan:dsna}}\id{mstmcc}%
For any fixed non-trivial minor-closed class~$\cal C$ of graphs, the Contractive Bor\o{u}vka's
algorithm (\ref{contbor}) finds the MST of any graph of this class in time
$\O(n)$. (The constant hidden in the~$\O$ depends on the class.)
\figure{hexangle.eps}{\epsfxsize}{The construction from Remark~\ref{hexa}}
\rem
-The observation in~Theorem~\ref{mstmcc} was also independently made by Gustedt \cite{gustedt:parallel},
-who studied a~parallel version of the Contractive Bor\o{u}vka's algorithm applied
+The observation in~Theorem~\ref{mstmcc} was also used by Gustedt \cite{gustedt:parallel},
+to construct parallel version of the Contractive Bor\o{u}vka's algorithm applied
to minor-closed classes.
\rem
\algin A~graph~$G$ with an edge comparison oracle.
\:$T\=\emptyset$. \cmt{edges of the MST}
\:$\ell(e)\=e$ for all edges~$e$. \cmt{edge labels as usually}
-\:$m_0\=m$.
+\:$m_0\=m$. \cmt{in the following, $n$ and $m$ will change with the graph}
\:While $n>1$: \cmt{We will call iterations of this loop \df{phases}.}
\::$F\=\emptyset$. \cmt{forest built in the current phase}
\::$t\=2^{\lceil 2m_0/n \rceil}$. \cmt{the limit on heap size}
The Iterated Jarn\'\i{}k's algorithm runs in time $\O(m\log^* n)$.
\proof
-$\beta(m,n) \le \beta(1,n) \le \log^* n$.
+$\beta(m,n) \le \beta(n,n) \le \log^* n$.
\qed
\cor\id{ijdens}%
flips in step~2 of this algorithm. We also know that the algorithm stops before
it adds $n$~edges to~$F$. Therefore it flips at most as many coins as a~simple
random process that repeatedly flips until it gets~$n$ heads. As waiting for
-every occurrence of heads takes expected time~$1/p$, waiting for~$n$ heads
-must take $n/p$. This is the bound we wanted to achieve.
+every occurrence of heads takes expected time~$1/p$ (the distribution is geometric),
+waiting for~$n$ heads must take $n/p$. This is the bound we wanted to achieve.
\qed
\para