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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. 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. 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. $A$ and $B$ exists, even though the curves do not bisect in the concrete plane.
$$ A \cap B \neq \emptyset $$ $$ A \cap B \neq \emptyset $$
@ -163,7 +163,11 @@ $$ A {\enc} C $$
As the relationships {\em enclosure} and {\pic} are mutually exclusive As the relationships {\em enclosure} and {\pic} are mutually exclusive
and {\em enclosure} is transitive and {\pic} is not, we can represent and {\em enclosure} is transitive and {\pic} is not, we can represent
an {\em enclosure} relationship as a directed vertice and 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] \pagebreak[1]
\section{The {\pic}} \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}. we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a}.
\begin{figure}[h] \begin{figure}[h]
\centering \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 % 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}.} \caption{Graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
\label{fig:eulerg_pic_g_a} \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] \begin{figure}[h]
\centering \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 % 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}.} \caption{Uncluttered graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
\label{fig:eulerg_pic_g_a_unc} \label{fig:eulerg_pic_g_a_unc}
\end{figure} \end{figure}
\pagebreak[1] %\pagebreak[9]
\clearpage
\section{Reduction of searches \\ for available zones} \section{Reduction of searches \\ for available zones}
Another property of any {\pic} $P$, is that 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 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}. 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. 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} \subsection{Available Zone Searching}
The available zones in an Euler diagram represent set theoretic combinations The available zones in an Euler diagram represent set theoretic combinations
that can be used in the diagram. that can be used in the diagram.
%For FMMD analyis, the test~cases %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 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). 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 Using $|P|$ to represent the number of conoutrs within a {\pic}
(the {\pic}s) which have an order $|P|.2^{|P|}$. 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.} 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 $|P|.2^{|P|}$ reduced by requiring several $K.|P|.2^{|P|}$
instead of $N.2^N$ as $P \leq N$. instead of $N.2^N$ as $K < N$.
\vspace{40pt} \vspace{40pt}