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Euler diagrams as graphs doc tidied

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Robin Clark 2010-08-21 13:15:20 +01:00
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commit 9b30d7d570
1 changed files with 31 additions and 12 deletions
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eulerg/eulerg.tex
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@ -104,9 +104,11 @@ $$ D \cap E \neq \emptyset $$
Clearly though these intersections are different, because
in the $A$, $B$ case
$$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $$
$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $
This is not the case for $D$, $E$ where:
$$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $$
$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $.
Another way of expressing this is that $A \cap B \neq \emptyset$ and
$ D \subset E$.
\paragraph{Enclosure}
To distinguish between these we can term the $A$, $B$ case to be
@ -194,15 +196,17 @@ Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and
\label{fig:eulerg_pic}
\end{figure}
\textbf{rule:}
%\textbf{rule:}
\begin{definition}
If any contour in a {\pic} is enclosed by any contour not belonging to the chain,
all the countours within the
{\pic} will be enclosed by it. This is because a contour
enclosing which bisects(????) another contour in a {\pic}
becomes part of the pure~intersection~chain. Hmmmm thats true but a better way to say it ????
{\pic} will be enclosed by it.
\end{definition}
This is because a contour
enclosing which bisects another contour in a {\pic}
becomes part of the pure~intersection~chain.
% Hmmmm thats true but a better way to say it ????
%The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.
@ -284,9 +288,8 @@ see figure \ref{fig:eulerg_pic_g_a_unc}.
\end{figure}
\pagebreak[4]
\section{reduction of searches for available zones}
\pagebreak[0]
\section{Reduction of searches for available zones}
Another property of any {\pic} $P$, is that
the maximum number of euler zones within it is
@ -300,12 +303,28 @@ a contour, we change the nature of the zones within
the {\pic}, but the number of zones contributed by the {\pic}
stays the same.
\begin{definition}
A pure intersection chain has a maximum number of possible Euler zones, and exists as independent entities in the diagram. This
allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.
\label{picreduction}
The number of available zones within a {\pic} $P$ does not change
when other contours are added or removed from the diagram
that are not, or would not become members of the {\pic} $P$.
\end{definition}
This is to say, the the number of zones within a {\pic} is not affected by changes in the diagram
his 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.
\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
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 $|P|.2^{|P|}$.
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 $P \leq N$.
\vspace{40pt}