From: Martin Mares <mj@ucw.cz>
Date: Sun, 4 May 2008 16:56:11 +0000 (+0200)
Subject: Noted that all logs are binary.
X-Git-Tag: printed~4
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=cd5388de13e5fe2390bd7ef3ea62318191d2b45d;p=saga.git

Noted that all logs are binary.
---

diff --git a/PLAN b/PLAN
index 8ca7a16..51732fb 100644
--- a/PLAN
+++ b/PLAN
@@ -63,7 +63,3 @@ Ranking:
 Typography:
 
 - formatting of multi-line \algin, \algout
-
-Global:
-
-- intro: \log is binary
diff --git a/adv.tex b/adv.tex
index 2bc916f..d56f58c 100644
--- a/adv.tex
+++ b/adv.tex
@@ -352,7 +352,7 @@ thus by the previous theorem the operations take $\O(m+n\log n)$ time in total.
 \qed
 
 \cor
-For graphs with edge density at least $\log n$, this algorithm runs in linear time.
+For graphs with edge density $\Omega(\log n)$, this algorithm runs in linear time.
 
 \remn{Other heaps}%
 We can consider using other kinds of heaps that have the property that inserts
@@ -723,7 +723,7 @@ We will calculate the number of comparisons~$c_i$ performed when processing the
 going from the $(i+1)$-th to the $i$-th level of the tree.
 The levels are numbered from the bottom, so leaves are at level~0 and the root
 is at level $\ell\le \lceil \log_2 n\rceil$. There are $n_i\le n/2^i$ vertices
-at the $i$-th level, so we consider exactly $n_i$ edges. To avoid taking a~logarithm
+at the $i$-th level, so we consider exactly $n_i$ edges. To avoid taking a~logarithm\foot{All logarithms are binary.}
 of zero, we define $\vert T_e\vert=1$ for $T_e=\emptyset$.
 \def\eqalign#1{\null\,\vcenter{\openup\jot
   \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil