From: Martin Mares <mj@ucw.cz>
Date: Sat, 30 Aug 2008 12:27:38 +0000 (+0200)
Subject: Fixes to Chapter 1.
X-Git-Tag: v2~6
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=5146864313f46256dbd7da9f41f27e1c587af108;p=saga.git

Fixes to Chapter 1.
---

diff --git a/TODO b/TODO
index ac91e76..1c719d8 100644
--- a/TODO
+++ b/TODO
@@ -17,10 +17,6 @@ Typography:
 
 Diaz:
 
-- 1.1.2: progress was made not only by means of new algorithms, but also
-  due to better data structures
-- 1.4.2: full list of authors
-- 1.4.20: proof -> comment
 - 2: simplify, remove proof sketches
 - 2.3: mention radix sorting in linear time
 - 2.5.24: skip proof
@@ -44,6 +40,7 @@ Patrice:
   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
   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
diff --git a/biblio.bib b/biblio.bib
index 867de5b..acb1f8c 100644
--- a/biblio.bib
+++ b/biblio.bib
@@ -1560,3 +1560,12 @@
     number = "2007/5",
     type = "Tech Report"
 }
+
+@article{ florek:liaison,
+  title={{Sur la liaison et la division des points d'un ensemble fini}},
+  author={Florek, K. and {\L}ukaszewicz, J. and Perkal, H. and Steinhaus, H. and Zubrzycki, S.},
+  journal={Colloquium Mathematicum},
+  volume={2},
+  pages={282--285},
+  year={1951}
+}
diff --git a/mst.tex b/mst.tex
index 5cccbb9..e0f4c46 100644
--- a/mst.tex
+++ b/mst.tex
@@ -45,7 +45,8 @@ In the next 50 years, several significantly faster algorithms were discovered, r
 from the $\O(m\timesbeta(m,n))$ time algorithm by Fredman and Tarjan \cite{ft:fibonacci},
 over algorithms with inverse-Ackermann type complexity by Chazelle \cite{chazelle:ackermann}
 and Pettie \cite{pettie:ackermann}, to an~algorithm by Pettie \cite{pettie:optimal}
-whose time complexity is provably optimal.
+whose time complexity is provably optimal. Frequently, the most important ingredients
+were advances in data structures used to represent the graph.
 
 In the upcoming chapters, we will explore this colorful universe of MST algorithms.
 We will meet the canonical works of the classics, the clever ideas of their successors,
@@ -372,7 +373,7 @@ edge of those having exactly one endpoint in the tree (we will call such edges
 the \df{neighboring edges} of the tree). We add all such edges to the forest and
 proceed with the next iteration.
 
-\algn{Bor\o{u}vka \cite{boruvka:ojistem}, Choquet \cite{choquet:mst}, Sollin \cite{sollin:mst}, and others}
+\algn{Bor\o{u}vka \cite{boruvka:ojistem}, Choquet \cite{choquet:mst}, Florek et al.~\cite{florek:liaison}, Sollin \cite{sollin:mst}}
 \algo
 \algin A~graph~$G$ with an edge comparison oracle.
 \:$T\=$ a forest consisting of vertices of~$G$ and no edges.
@@ -530,7 +531,7 @@ to add an~edge~$uv$, we first call $\<Find>(u,v)$ to check if both endpoints of
 the same component. If they do not, addition of this edge connects both components into one,
 so we perform $\<Union>(u,v)$ to merge the equivalence classes.
 
-Tarjan has shown that there is a~data structure for the DSU problem
+Tarjan \cite{tarjan:setunion} has shown that there is a~data structure for the DSU problem
 of surprising efficiency:
 
 \thmn{Disjoint Set Union, Tarjan \cite{tarjan:setunion}}\id{dfu}%
@@ -538,12 +539,9 @@ Starting with a~trivial equivalence with single-element classes, a~sequence of o
 comprising of $n$~\<Union>s intermixed with $m\ge n$~\<Find>s can be processed in time
 $\O(m\timesalpha(m,n))$, where $\alpha(m,n)$ is a~certain inverse of the Ackermann's function
 (see Definition \ref{ackerinv}).
-
-\proof
-See \cite{tarjan:setunion}.
 \qed
 
-This completes the following theorem:
+Using this data structure, we get the following bound:
 
 \thm\id{kruskal}%
 The Kruskal's algorithm finds the MST of a given graph in time $\O(m\log n)$.