Corrections of errors mentioned by Patrice.

This includes cleanup of light/heavy edges in presence of non-unique
weights.

diff --git a/Makefile b/Makefile
index e387ee03f7f5ccc5f7fb61dd279c555a5a981e5a..bb97dd00b32def16ffb4d8cac3916d6553e71eca 100644 (file)
--- a/Makefile
+++ b/Makefile
@@ -1,6 +1,6 @@
 all: saga.ps
 
-CHAPTERS=cover mst ram adv opt dyn rank notation
+CHAPTERS=cover intro mst ram adv opt dyn rank concl notation
 
 %.dvi: %.tex macros.tex biblio.bib
        tex $< && bibtex $* && tex $< && tex $<

diff --git a/PLAN b/PLAN
index f351b240fe936caf31d403ce50eff138b730f520..c768f978ed09a7724b89c2d631a7c3410c1c2ee5 100644 (file)
--- a/PLAN
+++ b/PLAN
@@ -48,6 +48,11 @@
 
 TODO:
 
+Preface:
+
+- move TOC to the beginning of the book
+- mention notation
+
 Spanning trees:
 
 - cite Eisner's tutorial \cite{eisner:tutorial}
@@ -68,6 +73,7 @@ Related:
 - K best trees
 - degree-restricted cases and arborescences
 - bounded expansion classes?
+- finding all MST's
 
 Ranking:
 
@@ -81,6 +87,7 @@ Ranking:
 
 Notation:
 
+- sort the table
 - G has to be connected, so m=O(n)
 - impedance mismatch in terminology: contraction of G along e vs. contraction of e.
 - use \delta(X) notation

diff --git a/adv.tex b/adv.tex
index 105152596f09bf8a62562fd16b6c6518186c8da7..12c634677f84d6d312b61d2c355e3a4aeda097ec 100644 (file)
--- a/adv.tex
+++ b/adv.tex
@@ -1080,7 +1080,9 @@ we will assume that we are working on the RAM.
 \lemman{Random sampling, Karger \cite{karger:sampling}}
 Let $H$~be a~subgraph of~$G$ obtained by including each edge independently
 with probability~$p$ and $F$~the minimum spanning forest of~$H$. Then the
-expected number of $F$-nonheavy edges in~$G$ is at most $n/p$.
+expected number of $F$-nonheavy\foot{These include $F$-light edges and also
+edges of~$F$ itself.}
+edges in~$G$ is at most $n/p$.
 
 \proof
 Let us observe that we can obtain the forest~$F$ by running the Kruskal's algorithm

diff --git a/intro.tex b/intro.tex
index b17d74a9d6d10cf61507b659c94d32a7f8916310..b3abdc474f0f6c0143b4638ae676854c29434dee 100644 (file)
--- a/intro.tex
+++ b/intro.tex
@@ -2,6 +2,6 @@
 \input macros.tex
 \fi
 
-\chapter{Introduction}
+\chapter{Preface}
 
 \endpart

diff --git a/mst.tex b/mst.tex
index 3f715954dd69cdaeb575751369442f8f562971f9..820ba51afd3bd4be0aa95a636f6059983ba1095d 100644 (file)
--- a/mst.tex
+++ b/mst.tex
@@ -24,7 +24,7 @@ find its minimum spanning tree, defined as follows:
 For a given graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$:
 \itemize\ibull
 \:A~subgraph $H\subseteq G$ is called a \df{spanning subgraph} if $V(H)=V(G)$.
-\:A~\df{spanning tree} of $G$ is any its spanning subgraph that is a tree.
+\:A~\df{spanning tree} of~$G$ is any spanning subgraph of~$G$ that is a tree.
 \:For any subgraph $H\subseteq G$ we define its \df{weight} $w(H):=\sum_{e\in E(H)} w(e)$.
   When comparing two weights, we will use the terms \df{lighter} and \df{heavier} in the
   obvious sense.
@@ -50,15 +50,16 @@ whose time complexity is provably optimal.
 In the upcoming chapters, we will explore this colorful universe of MST algorithms.
 We will meet the standard works of the classics, the clever ideas of their successors,
 various approaches to the problem including randomization and solving of important
-special cases.
+special cases. At several places, we will try to contribute our little stones to this
+mosaic.
 
 %--------------------------------------------------------------------------------
 
 \section{Basic properties}\id{mstbasics}%
 
 In this section, we will examine the basic properties of spanning trees and prove
-several important theorems to base the algorithms upon. We will follow the theory
-developed by Tarjan in~\cite{tarjan:dsna}.
+several important theorems which will serve as a~foundation for our MST algorithms.
+We will mostly follow the theory developed by Tarjan in~\cite{tarjan:dsna}.
 
 For the whole section, we will fix a~connected graph~$G$ with edge weights~$w$ and all
 other graphs will be spanning subgraphs of~$G$. We will use the same notation
@@ -80,9 +81,9 @@ Let~$T$ be a~spanning tree. Then:
 \endlist
 
 \rem
-Edges of the tree~$T$ itself are neither heavy nor light. We will sometimes
-use the name \df{non-heavy} for edges which are either light or contained
-in the tree.
+Edges of the tree~$T$ cover only themselves and thus they are neither heavy nor light.
+The same can happen if an~edge outside~$T$ covers only edges of the same weight,
+but this will be rare because all edge weights will be usually distinct.
 
 \lemman{Light edges}\id{lightlemma}%
 Let $T$ be a spanning tree. If there exists a $T$-light edge, then~$T$
@@ -126,7 +127,7 @@ hypothesis to $T^*$ and $T'$ to get the rest of the exchange sequence.
 \lemman{Monotone exchanges}\id{monoxchg}%
 Let $T$ be a spanning tree such that there are no $T$-light edges and $T'$
 be an arbitrary spanning tree. Then there exists a sequence of edge exchanges
-transforming $T$ to~$T'$ such that the weight does not increase in any step.
+transforming $T$ to~$T'$ such that the weight does not decrease in any step.
 
 \proof
 We improve the argument from the previous proof, refining the induction step.
@@ -421,7 +422,7 @@ by the lightest neighboring edge until it spans the whole graph.
 \endalgo
 
 \lemma
-Jarn\'\i{}k's algorithm computers the MST of the input graph.
+Jarn\'\i{}k's algorithm computes the MST of the input graph.
 
 \proof
 If~$G$ is connected, the algorithm always stops. Let us prove that in every step of