From ba1aa9941cd31290875dbe08ccd3ad2d81cee190 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Mon, 28 Jan 2008 11:58:57 +0100
Subject: [PATCH] Remark on edge densities.

---
 mst.tex      | 8 ++++++++
 notation.tex | 1 +
 2 files changed, 9 insertions(+)

diff --git a/mst.tex b/mst.tex
index dadd2a0..9c21ffa 100644
--- a/mst.tex
+++ b/mst.tex
@@ -1079,7 +1079,15 @@ The Iterated Jarn\'\i{}k's algorithm runs in time $\O(m\log^* n)$.
 $\beta(m,n) \le \beta(1,n) = \log^* n$.
 \qed
 
+\cor
+When we use the Iterated Jarn\'\i{}k's algorithm on graphs with edge density
+at least~$\log^{(k)} n$ for some $k\in{\bb N}^+$, it runs in time~$\O(km)$.
+
+\proof
+If $m/n \ge \log^{(k)} n$, then $\beta(m,n)\le k$.
+\qed
 
+\FIXME{Reference to Q-Heaps.}
 
 
 
diff --git a/notation.tex b/notation.tex
index 12cf520..3c63ac2 100644
--- a/notation.tex
+++ b/notation.tex
@@ -9,6 +9,7 @@
 \def\[#1]{[\ref{#1}]}
 \n{$\bb R$}{the set of all real numbers}
 \n{$\bb N$}{the set of all natural numbers, including 0}
+\n{${\bb N}^+$}{the set of all positive integers}
 \n{$T[u,v]$}{the path in a tree~$T$ joining vertices $u$ and $v$ \[heavy]}
 \n{$T[e]$}{the path in a tree~$T$ joining the endpoints of an~edge~$e$ \[heavy]}
 \n{$A\symdiff B$}{symetric difference of sets: $(A\setminus B) \cup (B\setminus A)$}
-- 
2.39.2