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eulerg

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Robin Clark 2010-08-31 07:25:12 +01:00
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eulerg/eulerg.tex
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@ -94,7 +94,7 @@ $A$ `enclosed' by $B$. We can express this as a directed relationship.
$$ B {\enc} A $$ $$ B {\enc} A $$
%\clearpage
\paragraph{Pair-wise Intersection} \paragraph{Pair-wise Intersection}
In the $D$, $E$ case we have In the $D$, $E$ case we have
@ -117,7 +117,7 @@ then B encloses C, see figure \ref{fig:eulerg_enc}.
No contour can be both pair-wisely intersected and enclosed with the same contour. No contour can be both pair-wisely intersected and enclosed with the same contour.
\end{definition} \end{definition}
\clearpage
\begin{figure}[h] \begin{figure}[h]
\centering \centering
@ -164,9 +164,9 @@ 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 on the same graph. {\pic} as non-directed on the same graph.
Figures \ref{fig:eulerg1} and \ref{fig:eulergenc} show euler diagrams with corresponding Figures \ref{fig:eulerg1} and \ref{fig:eulerg_enc} show euler diagrams with corresponding
graphs. The next section will introduce the concept of a {\pic} graphs. The next section will introduce the concept of a {\pic}
and will describe graphs where both enclosure and pair-wise and will present graphs where both enclosure and pair-wise
intersection are represented on the same graph. intersection are represented on the same graph.
\pagebreak[1] \pagebreak[1]
@ -321,7 +321,7 @@ Searching for an available zone involves finding out if the intersection exists,
by any other contours. by any other contours.
A brute force search for available zones using area operations 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).
Using $|P|$ to represent the number of conoutrs within a {\pic} Using $|P|$ to represent the number of contours within a {\pic}
and $K$ to represent the number of {\pic}s in a diagram, 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 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 {\pic}s) which have an order $K.2^{|P|}$.