From: Martin Mares <mj@ucw.cz>
Date: Sun, 3 Feb 2008 20:19:21 +0000 (+0100)
Subject: Fix definition of \beta and \log^*.
X-Git-Tag: printed~245
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=dbe915aea3c8d424f8bdca570f59bb50ac5b44b1;p=saga.git

Fix definition of \beta and \log^*.
---

diff --git a/adv.tex b/adv.tex
index 155bd98..400b8a0 100644
--- a/adv.tex
+++ b/adv.tex
@@ -376,7 +376,7 @@ how the algorithm could have stopped growing the tree~$R_i^j$:
 
 \thm\id{itjarthm}%
 The Iterated Jarn\'\i{}k's algorithm finds the MST of the input graph in time
-$\O(m\timesbeta(m,n))$, where $\beta(m,n):=\min\{ i: \log^{(i)}n < m/n \}$.
+$\O(m\timesbeta(m,n))$, where $\beta(m,n):=\min\{ i: \log^{(i)}n \le m/n \}$.
 
 \proof
 Phases are finite and in every phase at least one edge is contracted, so the outer
diff --git a/notation.tex b/notation.tex
index 80ebe3f..5517660 100644
--- a/notation.tex
+++ b/notation.tex
@@ -39,8 +39,8 @@
 \n{$\log n$}{a binary logarithm of the number~$n$}
 \n{$f^{(i)}$}{function~$f$ iterated $i$~times: $f^{(0)}(x):=x$, $f^{(i+1)}(x):=f(f^{(i)}(x))$}
 \n{$2\tower n$}{the tower function (iterated exponential): $2\tower 0:=1$, $2\tower (n+1):=2^{2\tower n}$}
-\n{$\log^* n$}{the iterated logarithm: $\log^*n := \min\{i: \log^{(i)}n < 1\}$; the inverse of~$2\tower n$}
-\n{$\beta(m,n)$}{$\beta(m,n) := \min\{i: \log^{(i)}n < m/n \}$ \[itjarthm]}
+\n{$\log^* n$}{the iterated logarithm: $\log^*n := \min\{i: \log^{(i)}n \le 1\}$; the inverse of~$2\tower n$}
+\n{$\beta(m,n)$}{$\beta(m,n) := \min\{i: \log^{(i)}n \le m/n \}$ \[itjarthm]}
 \n{$W$}{word size of the RAM \[wordsize]}
 \n{$\(x)$}{number~$x\in{\bb N}$ written in binary \[bitnota]}
 \n{$\(x)_b$}{$\(x)$ zero-padded to exactly $b$ bits \[bitnota]}