From: Martin Mares <mj@ucw.cz>
Date: Sat, 30 Aug 2008 14:03:41 +0000 (+0200)
Subject: Minor fixes to chapters 3 and 6.
X-Git-Tag: v2~3
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=ff49cabd35d3743ff060887a7fbf7ff15306691b;p=saga.git

Minor fixes to chapters 3 and 6.
---

diff --git a/TODO b/TODO
index 27cb653..a76d1b8 100644
--- a/TODO
+++ b/TODO
@@ -18,28 +18,19 @@ Typography:
 Diaz:
 
 > 2: simplify, remove proof sketches
-- 3: intro: replace "efficient" by "linear"
-- 3.1.7, 3.1.10: skip proof
-- 3.3: do we really need the full Komlos's result?
-- 3.3.17: maxflow: shouldn't Karzanov be cited, too?
-- 3.5.1: does the first alg give any insight?
-- 3.5.1: mention geometric distribution
-- 5.4.6: could we give a simple proof?
-- 5: mention graphs with moving vertices?
-- 6: mention d-regular graphs
-- 6: two monographs on Euclidean MST
-	M. Steel: Probability theory and combinatorial optimization, SIAM 1997
-	J. Yukich: Probability theory of classical Euclidean optimization problems, Springer 1998
+> 3.1.7, 3.1.10: skip proof
+> 3.5.1: does the first alg give any insight?
+> 5: mention graphs with moving vertices?
 
 Patrice:
 
-- Remark on 2.5.1: polynomial time could be replaced by sub-exponential time.
+> Remark on 2.5.1: polynomial time could be replaced by sub-exponential time.
 - For 1.5.6, you should probably quote D. Cheriton and R.E. Tarjan.
   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.]
-- In 3.1.12 and 3.1.16, you should make explicit the dependence of the
+> 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
   considering a minor closed class or another does not change the
diff --git a/adv.tex b/adv.tex
index d56f58c..a5a1e13 100644
--- a/adv.tex
+++ b/adv.tex
@@ -8,7 +8,7 @@
 
 The contractive algorithm given in Section~\ref{contalg} has been found to perform
 well on planar graphs, but in the general case its time complexity was not linear.
-Can we find any broader class of graphs where this algorithm is still efficient?
+Can we find any broader class of graphs where this algorithm is still linear?
 The right context turns out to be the minor-closed graph classes, which are
 closed under contractions and have bounded density.
 
@@ -57,12 +57,11 @@ guarantees that we can always find a~finite set of forbidden minors:
 \thmn{Excluded minors, Robertson \& Seymour \cite{rs:wagner}}
 For every non-trivial minor-closed graph class~$\cal C$ there exists
 a~finite set~$\cal H$ of graphs such that ${\cal C}=\Forb({\cal H})$.
+\qed
 
-\proof
 This theorem has been proven in a~long series of papers on graph minors
 culminating with~\cite{rs:wagner}. See this paper and follow the references
 to the previous articles in the series.
-\qed
 
 \para
 For analysis of the contractive algorithm,
@@ -1128,8 +1127,8 @@ The number of $F$-nonheavy edges is therefore equal to the total number of the c
 flips in step~2 of this algorithm. We also know that the algorithm stops before
 it adds $n$~edges to~$F$. Therefore it flips at most as many coins as a~simple
 random process that repeatedly flips until it gets~$n$ heads. As waiting for
-every occurrence of heads takes expected time~$1/p$, waiting for~$n$ heads
-must take $n/p$. This is the bound we wanted to achieve.
+every occurrence of heads takes expected time~$1/p$ (the distribution is geometric),
+waiting for~$n$ heads must take $n/p$. This is the bound we wanted to achieve.
 \qed
 
 \para
diff --git a/appl.tex b/appl.tex
index 802198f..f9d59ab 100644
--- a/appl.tex
+++ b/appl.tex
@@ -103,6 +103,10 @@ based on the sweep-line technique and the Red rule. For other
 variations on the geometric MST, see Eppstein's survey paper
 \cite{eppstein:spanning}.
 
+There are also plentiful interesting results on expected properties of the
+Euclidean MST of various random point configurations. These are well covered
+by the monographs of Steele \cite{steele:ptco} and Yukich \cite{yukich:pteucl}.
+
 \paran{Steiner trees}
 The constraint that the segments in the previous example are allowed to touch
 each other only in the given points looks artificial and it is indeed uncommon in
diff --git a/biblio.bib b/biblio.bib
index acb1f8c..ad43a45 100644
--- a/biblio.bib
+++ b/biblio.bib
@@ -1569,3 +1569,22 @@
   pages={282--285},
   year={1951}
 }
+
+@book { steele:ptco,
+    author = "J. Michael Steele",
+    title = "{Probability Theory and Combinatorial Optimization}",
+    series = "{CMBS-NSF Regional Conference Series in Applied Mathematics}",
+    volume = 69,
+    year = "1987",
+    publisher = "Society for Industrial and Applied Mathematics"
+}
+
+@book{ yukich:pteucl,
+  title={{Probability Theory of Classical Euclidean Optimization Problems}},
+  author={Yukich, J.E.},
+  year={1998},
+  publisher={Springer},
+  publisher = {Springer Verlag},
+  volume={1675},
+  series={{Lecture Notes in Math}},
+}