From 307763644ab43b80b253640dbe6d5a5551b46360 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Thu, 17 Jan 2008 00:32:33 +0100
Subject: [PATCH] Further figures.

---
 macros.tex      |    3 +-
 mst.tex         |   14 +-
 pic/mst-bez.eps | 1572 ++++++++++++++++++++
 pic/mst-bez.vrr |   48 +
 pic/mst-rb.eps  | 3724 +++++++++++++++++++++++++++++++++++++++++++++++
 pic/mst-rb.vrr  |   92 ++
 6 files changed, 5447 insertions(+), 6 deletions(-)
 create mode 100644 pic/mst-bez.eps
 create mode 100644 pic/mst-bez.vrr
 create mode 100644 pic/mst-rb.eps
 create mode 100644 pic/mst-rb.vrr

diff --git a/macros.tex b/macros.tex
index 270d095..6085f87 100644
--- a/macros.tex
+++ b/macros.tex
@@ -271,7 +271,7 @@
 \seccount=0
 \thmcount=0
 
-\def\chapter#1{\vfill\eject
+\def\chapter#1{\vfill\supereject
 \advance\chapcount by 1
 \seccount=0
 \thmcount=0
@@ -350,6 +350,7 @@
 \medskip
 \centerline{\epsfxsize=#2\epsfbox{pic/#1}}
 \medskip
+\smallskip
 \centerline{#3}
 \endinsert}
 
diff --git a/mst.tex b/mst.tex
index ff08ec8..d61cecb 100644
--- a/mst.tex
+++ b/mst.tex
@@ -190,10 +190,10 @@ Most MST algorithms can be described as special cases of the following procedure
 \algn{Red-Blue Meta-Algorithm}
 \algo
 \algin A~graph $G$ with an edge comparison oracle (see \thmref{edgeoracle})
-\:In the beginning, all edges are uncolored.
+\:In the beginning, all edges are colored black.
 \:While possible, use one of the following rules:
-\::Pick a cut~$C$ such that its lightest edge is not blue and color it blue. \cmt{Blue rule}
-\::Pick a cycle~$C$ such that its heaviest edge is not red and color it red. \cmt{Red rule}
+\::Pick a cut~$C$ such that its lightest edge is not blue \hfil\break and color this edge blue. \cmt{Blue rule}
+\::Pick a cycle~$C$ such that its heaviest edge is not red \hfil\break and color this edge red. \cmt{Red rule}
 \algout Minimum spanning tree of~$G$ consisting of edges colored blue.
 \endalgo
 
@@ -216,12 +216,16 @@ When an edge is colored red in any step of the procedure, it is not contained in
 \proof
 \qed
 
-\lemman{Colorless lemma}
-When there exists an uncolored edge, then at least one rule can be applied.
+\figure{mst-rb.eps}{289pt}{Proof of the Blue (left) and Red (right) lemma}
+
+\lemman{Black lemma}
+As long as there exists a black edge, at least one rule can be applied.
 
 \proof
 \qed
 
+\figure{mst-bez.eps}{295pt}{Configurations in the proof of the Black lemma}
+
 \theoremn{Red-Blue correctness}
 For any selection of rules, the Red-Blue procedure stops and the blue edges form
 the minimum spanning tree of the input graph.
diff --git a/pic/mst-bez.eps b/pic/mst-bez.eps
new file mode 100644
index 0000000..cbffb6a
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+++ b/pic/mst-bez.eps
@@ -0,0 +1,1572 @@
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+%%BeginResource: font CMSY7
+%!PS-AdobeFont-1.1: CMSY7 1.0
+%%CreationDate: 1991 Aug 15 07:21:52
+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMSY7) readonly def
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+/Weight (Medium) readonly def
+/ItalicAngle -14.035 def
+/isFixedPitch false def
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+dup 122 /daggerdbl put
+dup 123 /paragraph put
+dup 124 /club put
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+
+% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.
+
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2.39.2