From: Martin Mares <mj@ucw.cz>
Date: Sat, 13 Sep 2008 13:41:44 +0000 (+0200)
Subject: Corrected bugs reported by Koubek.
X-Git-Tag: v2~2
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=124f193d14bed24539c5cd59407637cc38dbe740;p=saga.git

Corrected bugs reported by Koubek.
---

diff --git a/TODO b/TODO
index a76d1b8..cb7c13b 100644
--- a/TODO
+++ b/TODO
@@ -29,7 +29,7 @@ Patrice:
   Finding Minimum Spanning Trees. SIAM J. on Comp. 5(4) (1976) pp.
   724-742. who gave a linear time algorithm for planar graphs, extended by
   Tarjan in 1983 to proper minor closed classes (both quoted by Gustedt).
-  [XXX: Cannot get the paper.]
+  [XXX: The paper should be in the library at MS.]
 > In 3.1.12 and 3.1.16, you should make explicit the dependence of the
   running time  with respect, for instance, to the Hadwiger number of the
   graph or to the maximal density nabla(G) of a minor of the graph, as
diff --git a/adv.tex b/adv.tex
index a5a1e13..35d5d9b 100644
--- a/adv.tex
+++ b/adv.tex
@@ -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}%
diff --git a/mst.tex b/mst.tex
index e0f4c46..8a6a12a 100644
--- a/mst.tex
+++ b/mst.tex
@@ -328,8 +328,8 @@ and $V\setminus M$ contains no blue edges, therefore we can use the Blue rule.
 
 \nota\id{deltanota}%
 We will use $\delta(M)$ to denote the cut separating~$M$ from its complement.
-That is, $\delta(M) = E \cap (M \times (V\setminus M))$. We will also abbreviate
-$\delta(\{v\})$ as~$\delta(v)$.
+That is, $\delta(M) = \{ uv \in E \mid u\in M, v\not\in M \}$.
+We will also abbreviate $\delta(\{v\})$ as~$\delta(v)$.
 
 \thmn{Red-Blue correctness}%
 For any selection of rules, the Red-Blue procedure stops and the blue edges form
diff --git a/notation.tex b/notation.tex
index f8a512f..e02cccf 100644
--- a/notation.tex
+++ b/notation.tex
@@ -113,7 +113,7 @@ better define it carefully, following \cite{diestel:gt}:
 
 \defn A~\df{multigraph} is an ordered triple $(V,E,M)$, where $V$~is the
 set of vertices, $E$~is the set of edges, taken as abstract objects disjoint
-with the vertices, and $M$ is a mapping $E\rightarrow V \cup {V \choose 2}$
+with the vertices, and $M$ is a mapping $E\rightarrow {V \choose 2} \cup {V \choose 1}$
 which assigns to each edge either a pair of vertices or a single vertex
 (if the edge is a loop).
 
diff --git a/ram.tex b/ram.tex
index 095d54d..ac0a4ac 100644
--- a/ram.tex
+++ b/ram.tex
@@ -266,7 +266,7 @@ set of edges in the current graph and then flatten the graph, all this in time $
 We have spared the technical details for this section, in which we are going to
 explain several rather general techniques based on bucket sorting.
 
-As we have already suggested in the proof of Lemma \ref{contbor}, contractions
+As we have already suggested in the proof of Lemma \ref{contiter}, contractions
 can be performed in linear time by building an~auxiliary graph and finding its
 connected components. We will thus take care only of the subsequent flattening.