BUGS: Little ones to fix

[saga.git] / adv.tex

diff --git a/adv.tex b/adv.tex
index a5a1e13bf0cae8e6fe63af3fa0eab5a0edeafee5..6cab6b7b3abca6a149b8bf74cf8ca9382abf283c 100644 (file)
--- a/adv.tex
+++ b/adv.tex
@@ -73,7 +73,7 @@ theory.
 \defn\id{density}%
 Let $G$ be a~graph and $\cal C$ be a class of graphs. We define the \df{edge density}
 $\varrho(G)$ of~$G$ as the average number of edges per vertex, i.e., $m(G)/n(G)$. The
-edge density $\varrho(\cal C)$ of the class is then defined as the infimum of $\varrho(G)$ over all $G\in\cal C$.
+edge density $\varrho(\cal C)$ of the class is then defined as the supremum of $\varrho(G)$ over all $G\in\cal C$.
 
 \thmn{Mader \cite{mader:dens}}\id{maderthm}%
 For every $k\in{\bb N}$ there exists $h(k)\in{\bb R}$ such that every graph
@@ -125,7 +125,7 @@ $G$~would contain a~subdivision of~$K_x$ and hence $K_x$ as a~minor.
 
 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.)
@@ -256,8 +256,8 @@ has degree~9.
 \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
@@ -419,7 +419,7 @@ contract the edges of~this forest and iterate.
 \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}
@@ -504,7 +504,7 @@ time (Lemma~\ref{ijphase}).
 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}%