From 04c94902782997662564f77b59a63924659569af Mon Sep 17 00:00:00 2001
From: Robin Clark <robin48gx@gmail.com>
Date: Sun, 29 Aug 2010 13:26:08 +0100
Subject: [PATCH] .

---
 eulerg/eulerg.tex | 31 ++++++++++++++++++++-----------
 1 file changed, 20 insertions(+), 11 deletions(-)

diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex
index 8171af6..7c1216f 100644
--- a/eulerg/eulerg.tex
+++ b/eulerg/eulerg.tex
@@ -71,7 +71,7 @@ processing and parsing of the diagram without resorting to extra area operations
 
 The set theory term `intersection' can apply to both the curves overlapping and to the sub-set case.
 Intersection in a concrete diagram can mean two curves bisecting.
-For instance in diagram \ref{fig:eulerg1} the set theoretic intersection between
+For instance in figure \ref{fig:eulerg1} the set theoretic intersection between
 $A$ and $B$ exists, even though the curves do not bisect in the concrete plane.
 
 $$ A \cap B \neq \emptyset $$
@@ -163,7 +163,11 @@ $$ A {\enc} C $$
 As the relationships {\em enclosure} and {\pic} are mutually exclusive
 and {\em enclosure} is transitive and {\pic} is not, we can represent
 an {\em enclosure} relationship as a directed vertice and 
-{\pic} as non-directed.
+{\pic} as non-directed on the same graph.
+Figures \ref{fig:eulerg1} and \ref{fig:eulergenc} show euler diagrams with corresponding 
+graphs. The next section will introduce the concept of a {\pic}
+and will describe graphs where both enclosure and pair-wise
+intersection are represented on the same graph.
 
 \pagebreak[1]
 \section{The {\pic}}
@@ -257,7 +261,7 @@ Because enclosure is a directed relationship and {\em pair-wise intersection} is
 we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a}.
 \begin{figure}[h]
  \centering
- \includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
+ \includegraphics[width=300pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
  % eulerg_pic_g_a.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
  \caption{Graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
  \label{fig:eulerg_pic_g_a}
@@ -276,14 +280,15 @@ in order to show that contour A encloses all contours in $PIC1$.
 
 \begin{figure}[h]
  \centering
- \includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg}
+ \includegraphics[width=300pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg}
  % eulerg_pic_g_a_unc.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
  \caption{Uncluttered graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
  \label{fig:eulerg_pic_g_a_unc}
 \end{figure}
 
 
-\pagebreak[1]
+%\pagebreak[9]
+\clearpage
 \section{Reduction of searches \\ for available zones}
 
 Another property of any {\pic} $P$, is that
@@ -306,20 +311,24 @@ that are not, or would not become members of the {\pic} $P$.
 That is to say, the the number of zones within a {\pic} is not affected by changes in the diagram 
 that do not alter the {\pic}.
 This allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.
-
+\pagebreak[3]
 \subsection{Available Zone Searching}
 
 The available zones in an Euler diagram represent set theoretic combinations
 that can be used in the diagram. 
 %For FMMD analyis, the test~cases
 Searching for an available zone involves finding out if the intersection exists, and then determining whether it is covered up
-by any other contours. A brute force search for available zones using area operations
+by any other contours. 
+A brute force search for available zones using area operations
 is therefore of the order $N.2^N$ (where N is the number of contours in the diagram).
-By using the result in definition \ref{picreduction}, we can break the diagram into small segments
-(the {\pic}s) which have an order $|P|.2^{|P|}$.
+Using $|P|$ to represent the number of conoutrs within a {\pic} 
+and $K$ to represent the number of {\pic}s in a diagram,
+using the result in definition \ref{picreduction}, we can break the diagram into small segments
+(the {\pic}s) which have an order $K.2^{|P|}$.
+The exponential $2^N$ overhead is thus broken down into several smaller $2^{|P|}$ operations.
 The order of area operations is generally\footnote{In the case where the diagram is not comprised of just one {\pic}, which has no enclosing contours.} 
-reduced  by requiring several  $|P|.2^{|P|}$
-instead of $N.2^N$ as $P \leq N$.
+reduced  by requiring several  $K.|P|.2^{|P|}$
+instead of $N.2^N$ as $K < N$.
 
 \vspace{40pt}