From 1945e3fcfc7a8306ab351a35d20c6e746513c182 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Sat, 26 Jan 2008 00:03:11 +0100
Subject: [PATCH] More Fibonacci.

---
 macros.tex   |  6 ++++--
 mst.tex      | 15 +++++++++++++++
 notation.tex |  1 +
 3 files changed, 20 insertions(+), 2 deletions(-)

diff --git a/macros.tex b/macros.tex
index bedae0c..eb4d4ee 100644
--- a/macros.tex
+++ b/macros.tex
@@ -294,9 +294,11 @@
 \medskip
 }
 
-\def\proclaim#1{\advance\thmcount by 1
+\def\para{\advance\thmcount by 1
 \edef\currentid{\the\chapcount.\the\seccount.\the\thmcount}
-\noindent {\bo \currentid.\enspace #1.\enspace}}
+\noindent {\bo \currentid.\enspace}}
+
+\def\proclaim#1{\para {\bo #1.\enspace}}
 
 \def\thm{\proclaim{Theorem}}
 \def\lemma{\proclaim{Lemma}}
diff --git a/mst.tex b/mst.tex
index b8396b2..9bde5ab 100644
--- a/mst.tex
+++ b/mst.tex
@@ -595,6 +595,7 @@ We will also keep a list of all vertex structures. During the bucket sort, each
 structure will contain a pointer to the corresponding bucket and the vertex list will
 define the order of vertices (which can be arbitrary).
 
+\para
 Graph contractions are indeed a~very powerful tool and they can be used in other MST
 algorithms as well. The following lemma shows the gist:
 
@@ -944,8 +945,22 @@ authors claim implementation advantages.
 
 \FIXME{Mention Thorup's Fibonacci-like heaps for integers?}
 
+\rem
+For sparse graphs, we can cross-breed this algorithm with the contractive
+Bor\o{u}vka's algorithm: First perform $\log\log n$ steps of contractive
+Bor\o{u}vka, which takes $\O(m\log\log n)$ time and produces a~graph~$G'$
+with $m'\le m$ and $n'\le n/\log n$. Then run Algorithm~\ref{jarniktwo} on~$G'$
+and it finishes in time $\O(m'+n'\log n') = \O(m)$. Finally combine MST edges
+found by both algorithms to a~single tree as in~the Contraction lemma (\ref{contlemma}).
+
+\para Xyzzy.
+
+\FIXME{Describe the heap limitation trick and the $\O(m\beta(m,n))$ algorithm.}
+
+
 
 
+% use \para
 % G has to be connected, so m=O(n)
 % mention Steiner trees
 % mention matroids
diff --git a/notation.tex b/notation.tex
index 429e7dd..0da0035 100644
--- a/notation.tex
+++ b/notation.tex
@@ -35,6 +35,7 @@
 \n{$\deg_G(v)$}{degree of vertex~$v$ in graph~$G$; we omit $G$ if it is clear from context}
 \n{${\bb E}X$}{expected value of a~random variable~$X$}
 \n{${\rm Pr}[\varphi]$}{probability that a predicate~$\varphi$ is true}
+\n{$\log n$}{a binary logarithm of the number~$n$}
 }
 
 \section{Multigraphs and contractions}
-- 
2.39.5