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Robin Clark 2010-08-29 13:26:08 +01:00
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eulerg/eulerg.tex
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@ -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}