983 lines
38 KiB
TeX
983 lines
38 KiB
TeX
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\section{Introduction : Euler Diagram and Zones Available for use}
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Euler diagrams consist of closed curves in the plane which are used to represent sets.
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The spaitial ralationship between the curves defines the set theoretic relationships.
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\begin{itemize}
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\item Intersection - if the curves overlap
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\item Sub-set - if a curve is enclosed by another
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\item disjoint - if the curves are separate
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\end{itemize}
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A zone is defined as a region of the plane where a set of curves will enclose it
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and another set is disjoint from it.
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A Venn diagram is an Euler diagram where all possible zones are present.
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Thus if we have a Venn diagram with $N$ countours it will have $2^N$
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zones.
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%
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An Euler diagram, does not have to make all possible zones available.
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Thus we can make certain combinations of contours unavailable
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by drawing the diagrams. Or in other words we can deliberately restrict
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not only combinations of contours but also the number of available zones.
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%
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In Constraint Diagram and PLD's Euler diagrams are used
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and objects are placed upon the available zones.
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%
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When performing logical reasoning on euler diagrams,
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it is important to note which available zones do not have objects associated with them in order to flag the unhandled cases in the diagram.
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These could represent cases where the user has left them undefined, or considers them to
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be a general case. Either way they need to be flagged as an ommission error or
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collected. In order to do this a method of finding all available zones is necessary.
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%
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Also note that in a complicated diagram, a zone may, at first glance, appear available, but could be covered-up or obscured by
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an enclosing contour.
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%The sizes or shapes of the curves are not important, the significance of the diagram is in how they overlap.
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%The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness).
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%
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%Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements which are not members of the set.
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%Curves which do not intersect represent disjoint sets. Two curves which intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve which is contained completely within another represents a subset of it.
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%
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%Venn diagrams are a more restrictive form of Euler diagrams.
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%A Venn diagram must contain all the possible zones of overlap between its curves, representing all combinations of inclusion/exclusion of its constituent sets, but in an Euler diagram some zones might be missing.
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%Therefore there is only one Venn diagram representing the relationships between n sets, with 2n zones, but there may be many Euler diagrams.
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%(An example is given below in the History section; in the top-right illustration the O and I diagrams are merely rotated; Venn stated that this difficulty in part led him to develop his diagrams).
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\par
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For a zone to be available for use it must
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\begin{itemize}
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\item {Exist} meaning that the contours making up the zone must intersect.
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\item {Not be Obscured} The intersection must not be covered by other zones.
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\end{itemize}
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\par
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Each zone can be defined by the contours that intersect to define it and the remaining
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contours in the diagram.
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Thus for a diagram $ D $ consisting of a set of zones $ Z $ where the zones in the diagram
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$ Z_{n} $
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are defined as a sets of intersection contours $ A_{n} $, and exclusion sets
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$ B_{n} $.
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\subsubsection{Testing a Zone for Existance}
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The Java Area, Shape and Polygon classes, provide functions to
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intersect, subtract and `exclusive or' Areas on the plane. They are thus
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very useful in testing objects drawn under a Java environment.
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%
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To determine if a zone exists we can apply the intersection
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functions to all the $A_{N}$ contours. If any Area remains
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the zone exists. To check that the zone is available, we must
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also ensure that it is not covered over by a contour in the $ B_{N}$ set.
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\subsubsection{Testing for Obscuration}
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Firstly apply intersection to all the $A_{n}$ contours in the Zone.
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We are then left with the area of intersection only.
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%
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From this we can subtract all the areas from the $B_{n}$ contours.
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If the zone was obscured, or covered up the Area object will register that it has no area on the plane.
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%
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In other words, a zone obscured by other contours is one that forms the area of intersection
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of the set $ A_{n} $ and then having all the areas from set $ B_{n} $ subtracted from it
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no surface area left.
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%
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Firstly let us define the meaning of availability in concrete area termsi by means of an example diagram.
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In figure \ref{fig:ex1}, there is an Euler diagram with the following zones available.
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\par
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\vspace{0.3cm}
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\begin{tabular}{||l|l|c|c||} \hline \hline
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{\em Included set $A_{n}$ } & {\em Excluded $set B_{n}$ } & & {\em Available} \\ \hline
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\{ \} & \{ D C B A \} & 1 & Y \\ \hline
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\{ A \} & \{ D C B \} & 2 & Y \\ \hline
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\{ B \} & \{ D C A \} & 3 & Y \\ \hline
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\{ B A\} & \{ D C \} & 4 & Y \\ \hline
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\{ C \} & \{ D B A \} & 5 & N \\ \hline
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\{ C A \} & \{ D B \} & 6 & N \\ \hline
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\{ C B \} & \{ D A \} & 7 & N \\ \hline
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\{ C B A \} & \{ D \} & 8 & N \\ \hline
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\{ D \} & \{ C B A \} & 9 & Y \\ \hline
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\{ D A \} & \{ C B \} & 10 & N \\ \hline
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\{ D B \} & \{ C A \} & 11 & Y \\ \hline
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\{ D B A\} & \{ C \} & 12 & N \\ \hline
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\{ D C \} & \{ B A \} & 13 & Y \\ \hline
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\{ D C A \} & \{ B \} & 14 & N \\ \hline
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\{ D C B \} & \{ A \} & 15 & Y \\ \hline
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\{ D C B A \} & \{ \} & 16 & N \\ \hline
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\hline
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\end{tabular}
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\vspace{0.3cm}
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Note $B \cap C$ and $ C $ are not available in this diagram because it is impossible to place objects on them.
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Objects could be placed on $ B \cap C \cap D $ and $ B \cap D $ however.
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction1.jpg}
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% exampleareasubtraction1.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
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\caption{Simple Euler Diagram}
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\label{fig:ex1}
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\end{figure}
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%
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% \begin{figure}
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% \vskip 4.2cm
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% \special{psfile=fzd/exampleareasubtraction1.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
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% Simple Euler Diagram
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% \label{fig:ex1}}
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% \end{figure}
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Examining the intersection $ A \cap B $ the corresponding Area is shown in red
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in figure \ref{fig:ex2}.
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction2.jpg}
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% exampleareasubtraction2.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
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\caption{Area representation of intersection between A and B}
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\label{fig:ex2}
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\end{figure}
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% \begin{figure}
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% \vskip 4.2cm
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% \special{psfile=fzd/exampleareasubtraction2.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
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% Area representation of intersection between set A and B
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% \label{fig:ex2}}
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% \end{figure}
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The area to be subtracted is shown in blue in figure \ref{fig:ex3}.
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Note that here the intersection exists
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and is not obscured by the areas made up from the other contours.
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It is therefore available.
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It can be seen that the zone $ B \cap C $ is not available in the
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diagram because it is obscured by the
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areas comprised of the other contours $ C \cup D $
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(in fact in this diagram only $ D $ is required to prove obscuration
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because $ C \subset D $ ).
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction3.jpg}
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% exampleareasubtraction3.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
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\caption{Area representation of exclusion zone to be subtracted}
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\label{fig:ex3}
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\end{figure}
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%
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% \begin{figure}
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% \vskip 4cm
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% \special{psfile=fzd/exampleareasubtraction3.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
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% Area representation of exclusion Zone to be subtracted
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% \label{fig:ex3}}
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% \end{figure}
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%\begin{picture}(200,200)(10,10)
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%\put(20,0){\circle{20}}
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%\end{picture}
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%\clearpage
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\subsubsection{Formal expression of Area Operations}
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This section deals with operations on the concrete diagrams using Areas.
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An intersection for instance therefore represents an intersection of the Areas
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denoted by the contours.
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\equation
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\label{available5}
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{AreaIntersection}_{n} = \; \stackrel{\bigcap a }{a \in A_{n} \wedge a \not\in B_{n}}
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\endequation
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Note this implies contours in $B_{n}$ could intersect with the zone formed by $A_{n}$, but where all
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contours in $B_{n}$ represent those excluded from the zone under scrutinty.
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\par
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The $AreaIntersection_{n}$ may form a zonal region,% \cite{cried},
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but for the purposes of this investigation,
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may be interpreted as a boolean variable.
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There is either surface area or not. The zone exists or does not.
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\equation
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\label{available6}
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{AreaExclusion}_{n} = \; \stackrel{\bigcup b }{b \not\in A_{n} \wedge b \in B_{n} }
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\endequation
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%\begin{eqnarray}
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\equation
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\label{available7}
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\begin{array}[t]{c}
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{RemainingIntersection}_{n} = \\
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{AreaIntersection}_{n} - {AreaExclusion}_{n}
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\end{array}
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\endequation
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%\end{eqnarray}
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Note the subtraction here is again, an Area operation.
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The remaining intersection could be one or more visible zones on the diagram,
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but as long as there is some surface area
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left, then the zone has not been obscured.
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It can thus be used as a boolean variable. If $surface\_area > 0 $ exists in the ${RemainingIntersection}_{n}$ then
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the expression is true.
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%\clearpage
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\subsubsection{Testing for zone Availability}
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Firstly that the intersection exists in the concrete diagram.
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\equation
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\label{available1}
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%Z_{n} = (A_{n},B_{n}) \Rightarrow \forall \; A_{n} \; \mid \; a \cap A_{n}
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Z_{n} = (A_{n},B_{n}) \; \Rightarrow \; AreaIntersection_{n}
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\endequation
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If the above implication is true we can go on to test for obscuration.
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.
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\equation
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\label{available2}
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Z_{n} = (A_{n},B_{n}) \Rightarrow \; RemainingIntersection_{n}
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\endequation
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And finally the sets A and B added together must be equal to the set of all countours in
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diagram D, i.e. all contours must be represented in $A_{n}$ and $B_{n}$.
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\equation
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\label{available3}
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B_{n} \cup A_{n} = D
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\endequation
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A zone is thus considered {\em available} if the above criteria (i.e. equations \ref{available1},\ref{available2},\ref{available3} ) are true.
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\subsection {Algorthms for finding Available Zones}
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One could take all possible combinations of contours and check each case.
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For all combinations of contours, examine to find out if the intersection exists, and then if it does, examine all
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the excluded contours and ensure that they do not obscure the intersection. If both conditions are trus the zone is 'available'.
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\par
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Checking all possible combinations is henceforth referred to as the 'Brute force method'.
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\par
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The brute force method is simple, and
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would be practical for small numbers of contours. However as constraint/spider
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diagrams become used in practise larger numbers of contours and very large diagrams will become common.
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\par
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For instance were a diagram to contain 32 contours, to brute forcecheck for the existance of all contours would take
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all possible combinations of 32 objects. This corresponds to a binary count and thus $ 2^{32} $ possible zones to check for.
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\par
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A diagram with 32 contours would contain a potential of over 4 billion zones.
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\par
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Multiply that by the on average 16 operations ( area intersection compares ) that must be carried
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out for each zone, added to an average of 16 obscuration comparisons,
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and an astronomical number of area comparisons would be required ($ 32.2^{32} $).
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%A similar problem existed with computational frequency analysis
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%in the mid 1960's where the algorithmically more efficient FFT \cite{fft} became the industry standard
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%as opposed to the DFT.
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\par
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In general then the brute force zone search, with obscuration testing, takes
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\equation
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2^{N}.2.\frac{N}{2}
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\endequation
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or
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\equation
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\label{bruteforce}
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N.2^{N}
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\endequation
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\par
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This author therefore sees an efficient algorithm for determining available zones as a necessity
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for a practical euler/spider diagram analyser.
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\par
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The algorithm described in this paper works on the principle of examining the
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contours for two types of relationships.
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This information is then used to create directed graphs, which are then traversed. From the spanning trees,
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most of the non available zones are eliminated early on. For zones which may be available, two tests are required.
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The first that they actually exist in the concrete diagram, and secondly that they are not obscured by
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combinations of other contours.
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The initial searches form two cross products of the contours (see \ref{detpe} )
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performing $2.N^2$ area operations.
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After these initial area operations subsets of the contours can be defined (pure intersection chains)
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by traversing the pure intersection relationships. The pure intersection chains
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are subsets of the contours in the diagram.
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These subsets of the contours are denoted as the set $G$.
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In searching for the availability of zones, area operations only need be applied
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within each set $G$.
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By only having to search within subsets of the contours effeciency of the algorithm is improved.
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For the brute force algorithm, $ G \equiv N $.
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For each potential zone identified, it must be checked against $G\#$ other contours
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where $ G \subseteq N$
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for proof of concrete existence and
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for obscuration. The set $ G $ is divided into two, those included in the prospective zone $I$
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and those outside and
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used to check for obscuration $ O $ . Note $ O + I \leq N $ because $ G \subseteq N $.
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Thus for this algorithm, where $ Z\# $ is the number of potential zones as identified by tree spanning
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\equation
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\label{seteq}
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2.N^{2} + Z\#.G
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\endequation
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\equation
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\label{incobs}
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2.N^{2} + Z\#.(O + I)
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\endequation
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Thus for a $ Venn N $ diagram this algorithm is less efficient by $ 2.N^{2} $
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i.e. for a Venn N diagram $Z\# = N^{2}$
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\equation
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%\label{fzditer1}
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2.N^{2} + 2^{N}.2.\frac{N}{2}
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\endequation
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\equation
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%\label{fzditer1}
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2.N^{2} + 2^{N+1}
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\endequation
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But for any diagram less
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complicated than Venn N, where $ Z\# $
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is small in comparison with $2^{N}$ the algorithm becomes far more efficient.
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Examples of complexity savings are shown in section \ref{complexity}.
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 642 482,keepaspectratio=true]{fzd/piee.jpg}
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% piee.jpg: 891x669 pixel, 100dpi, 22.63x16.99 cm, bb=0 0 642 482
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\caption{A Pure Intersection and an Enclosure}
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\label{fig:piee}
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\end{figure}
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% \begin{figure}
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% \vskip 6cm
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% \special{psfile=fzd/piee.ps hoffset=0 voffset=-10 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
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% A Pure Intersection and an Enclosure
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% \label{fig:piee}}
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% \end{figure}
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\subsection { Relationships between Contours }
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The algorithm for fast finding of available zones depends upon defining three new relationships between contours.
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\begin{itemize}
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\item {Pure Intersection}
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\item {Enclosure}
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\item {Belonging to a Pure Intersection Chain}
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\end{itemize}
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\subsubsection { Pure Intersection }
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A pair of contours are said to have 'pure intersection' if the contours overlap.
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\subsubsection{Enclosure}
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A countour is said to be enclosed if it fits completely within another contour.
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\subsubsection { Pure Intersection Chains }
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 642 482]{fzd/pic.jpg}
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% pic.jpg: 891x669 pixel, 100dpi, 22.63x16.99 cm, bb=0 0 642 482
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\caption{Pure Intersection Zone Chain}
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\label{fig:pic}
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\end{figure}
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%
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% \begin{figure}
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% \vskip 6cm
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% \special{psfile=fzd/pic.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
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% Pure Intersection Zone Chain
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% \label{fig:pic}}
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% \end{figure}
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Pairs of contours may belong to the same pure intersection chain.
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Pure Intersection chains are a chain of contours that can all
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be linked together by pure intersection relationships ( see figure \ref{fig:pic} ).
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Contours in the pure intersection chain may enclose other members
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in the same chain, but not the contour that they are purely intersected with (see figure \ref{fig:picwaie}).
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\par
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Pure intersections relationships for the diagram in figure \ref{fig:pic} are :-
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$$
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\begin{array}{lcl}
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A & \stackrel{pi}\longrightarrow & B \\
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B & \stackrel{pi}\longrightarrow & A \\
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B & \stackrel{pi}\longrightarrow & C \\
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C & \stackrel{pi}\longrightarrow & B \\
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D & \stackrel{pi}\longrightarrow & C \\
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C & \stackrel{pi}\longrightarrow & D \\
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\end{array}
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$$
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\par
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Relationships for the diagram in figure \ref{fig:picwaie} include one enclosure.
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$$
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\begin{array}{lcl}
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A & \stackrel{pi}\longrightarrow & B \\
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B & \stackrel{pi}\longrightarrow & A \\
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B & \stackrel{pi}\longrightarrow & C \\
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C & \stackrel{pi}\longrightarrow & B \\
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D & \stackrel{pi}\longrightarrow & C \\
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C & \stackrel{pi}\longrightarrow & D \\
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E & \stackrel{pi}\longrightarrow & C \\
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C & \stackrel{pi}\longrightarrow & E \\
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\\
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E & \stackrel{enc}\longrightarrow & D \\
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\end{array}
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$$
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Because {\em A,B,C,D} all share pure intersection connections they all belong to the same pure intersection chain.
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NB: Three or more linked pure intersections contitute a 'pure intersection chain'.
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\subsubsection { Determining the Pure Intersection and Enclosure Relationships }
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\label{detpe}
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By applying java Area searches for enclosure and intersection on each
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contour against all others a collection of pure intersection relationships
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and enclosure relationships can be determined.
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This forms a list of relationship pairs from the cross product of all the contours.
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\equation
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%\label{crossprodsingle}
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\begin{array}{l}
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pi(a,b) \; \Rightarrow
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%\stackrel{\Delta}{=}
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\; \forall \; C \; \bullet \; a \; X \; \forall C \; \bullet \; b \\
|
|
\; \bullet (a \cap b) \; \wedge \; \neg (a \supseteq b) \; \wedge \; \neg (b \supseteq a) \\
|
|
\end{array}
|
|
\endequation
|
|
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 452 290,keepaspectratio=true]{fzd/pice.jpg}
|
|
% pice.jpg: 628x403 pixel, 100dpi, 15.95x10.24 cm, bb=0 0 452 290
|
|
\caption{Pure Intersection Chain with implicit enclosure}
|
|
\label{fig:picwaie}
|
|
\end{figure}
|
|
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/pice.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
|
|
% Pure Intersection Chain with an implicit Enclosure
|
|
% \label{fig:picwaie}}
|
|
% \end{figure}
|
|
|
|
\subsubsection {Determining Enclosure }
|
|
|
|
When a contour completely encloses another contour, it has an enclosing relationship.
|
|
See figure \ref{fig:piee}
|
|
|
|
To determine the enclosure relationships for the diagram,
|
|
all contours are checked for enclosure relationships with each other.
|
|
This again, forms a list of relationship pairs from cross product of all the contours.
|
|
|
|
|
|
\equation
|
|
%\label{crossprodsingle}
|
|
\begin{array}{l}
|
|
enc(a,b) \;
|
|
%\stackrel{\Delta}{=}
|
|
\Rightarrow \; \forall \; C \; \; \bullet a \; X \; \forall \; C \; \bullet \; b \\
|
|
\; \bullet (a \supset b) \\
|
|
\end{array}
|
|
\endequation
|
|
|
|
|
|
\subsection { Rules that can be derived from the three relationships }
|
|
|
|
\subsubsection { Rule 1: Simple Zone Creation }
|
|
|
|
Any contour not belonging to a pure intersection chain, will create a zone containing itself, and any enclosing contours.
|
|
|
|
\subsubsection { Rule 2: All Pure Intersection chains and enclosures can be represented on a directed graph }
|
|
|
|
By displaying pure intersection relations and enclosure relations in different colours
|
|
both can be represented on the same coloured directed graph (CDG). I have chosen blue for pure
|
|
intersections and red for enclosures for the examples that follow.
|
|
|
|
|
|
Figure \ref{fig:pig1} Shows a CDG for the diagram in figure \ref{fig:pic}. Note that traversing through this graph reveals all the intersections, and that there are no loops in the traversal, meaning that no multiple intersections exist.
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 361 163]{fzd/pig1.jpg}
|
|
% pig1.jpg: 502x227 pixel, 100dpi, 12.75x5.77 cm, bb=0 0 361 163
|
|
\caption{Coloured Directed Graph of Pure Intersection Chain}
|
|
\label{fig:pig1}
|
|
\end{figure}
|
|
%
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/pig1.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
|
|
% Coloured Directed Graph of Pure Intersection Chain
|
|
% \label{fig:pig1}}
|
|
% \end{figure}
|
|
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 352 307,keepaspectratio=true]{fzd/pig2.jpg}
|
|
% pig2.jpg: 489x427 pixel, 100dpi, 12.42x10.85 cm, bb=0 0 352 307
|
|
\caption{Coloured Directed Graph of Pure Intersection Chain with an implicit enclosure}
|
|
\label{fig:pig2}
|
|
\end{figure}
|
|
%
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/pig2.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
|
|
% Coloured Directed Graph of Pure Intersection Chain with an implicit Enclosure
|
|
% \label{fig:pig2}}
|
|
% \end{figure}
|
|
|
|
|
|
\subsubsection { Graph Traversal }
|
|
|
|
By traversing the graphs and applying tests for implicit enclosure within a pure intersection chain
|
|
from each contour belonging to it, and applying any enclosure relations all possible
|
|
zone combinations are revealed.
|
|
A path may not loop, i.e. it cannot branch to a contour all ready examined in the path.
|
|
|
|
\subsubsection{ Rule 3: Traversal Reduction : Avoiding Repeated Area checking }
|
|
As each potential zone is discovered and checked it is temporarily stored,
|
|
and if re-discovered on a new path, is not subjected to Area testing.
|
|
|
|
\subsubsection { Rule 4: Pure Intersection Pair Zone Creation }
|
|
|
|
If any pure intersection exists, a potential zone exists. This zone
|
|
will intersect with any contour which has an enclosure relationship with
|
|
any of the pair in the pure intersection.
|
|
It may also be enclosed by a member of the pure intersection chain it belongs to.
|
|
Thus where a pair intersection exists, checking for its existence is unnecessary,
|
|
but checking for obscuration is.
|
|
\par
|
|
Note by only searching within the pure intersection chain, unnecessary obscuration tests are avoided
|
|
and this refines the obscuration test in equation \ref{available7}.
|
|
\par
|
|
If a zone does not belong to a pure intersection chain, no obscuration testing is necessary.
|
|
\par
|
|
Note : In figure \ref{fig:picwaie} note that contours \em{D} and \em{E} are in
|
|
the same pure intersection chain, but that
|
|
\em{E} also encloses \em{D}.
|
|
|
|
|
|
|
|
|
|
\subsubsection { Rule 5: Multiple Zone Creation }
|
|
|
|
|
|
A circular reference (often described as a circuit \cite{gtl} \cite{alggraph}) containing more than one pair of pure intersections
|
|
indicates the possibility
|
|
of a multiple intersection of the contours in the path. The converse is true.
|
|
Should there be no circular path, there can be
|
|
multiple intersection along this path.
|
|
\par
|
|
This is best shown as an example, see figures \ref{fig:abc1}, \ref{fig:abc2} and \ref{fig:abc3}.
|
|
Consider the three contours A, B and C.
|
|
These belong to the same pure intersection chain,
|
|
and the graph has a circular reference. However one contains a an $ A \cap B \cap C $
|
|
zone and the other does not.
|
|
\par
|
|
Multiple intersections due to enclosure are discovered by traversing the
|
|
enclosure relations.
|
|
\par
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 375 377,keepaspectratio=true]{fzd/abc1.jpg}
|
|
% abc1.jpg: 521x524 pixel, 100dpi, 13.23x13.31 cm, bb=0 0 375 377
|
|
\caption{circular Reference with multiple zone}
|
|
\label{fig:abc1}
|
|
\end{figure}
|
|
%
|
|
% \label{fzd}
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/abc1.eps hoffset=0 voffset=0 hscale=60 vscale=60}\caption[Pure Intersection Zone Capture Method]{
|
|
% Circular reference with multiple zone
|
|
% \label{fig:abc1}}
|
|
% \end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 390 371]{fzd/abc2.jpg}
|
|
% abc2.jpg: 542x515 pixel, 100dpi, 13.77x13.08 cm, bb=0 0 390 371
|
|
\caption{Circular reference without multiple zone}
|
|
\label{fig:abc2}
|
|
\end{figure}
|
|
%
|
|
% \label{fzd}
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/abc2.eps hoffset=0 voffset=-20 hscale=60 vscale=60}\caption[Pure Intersection Zone Capture Method]{
|
|
% Circular reference without multiple zone
|
|
% \label{fig:abc2}}
|
|
% \end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 642 482,keepaspectratio=true]{fzd/abcgraph.jpg}
|
|
% abcgraph.jpg: 891x669 pixel, 100dpi, 22.63x16.99 cm, bb=0 0 642 482
|
|
\caption{Pure Intersection Zone and Enclosure}
|
|
\label{fig:abdgraph}
|
|
\end{figure}
|
|
%
|
|
% \label{fzd}
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/abcgraph.ps hoffset=0 voffset=-10 hscale=30 vscale=30}\caption[Pure Intersection Zone Capture Method]{
|
|
% Pure Intersection Zone and Enclosure
|
|
% \label{fig:abcgraph}}
|
|
% \end{figure}
|
|
|
|
In order to examine multiple intersections, the spanning tree from the contour under inspection must be
|
|
recursively iterated (only within the pure intersection chain it belongs to i.e. within the subset $G$).
|
|
|
|
Should a path arrive back at its start vertex, a circular reference has been found
|
|
and there is the possibility of a multiple intersection zone, see figure \ref{fig:abcgraph}.
|
|
\par
|
|
Note that the contour maps for {A,B,C} in figure \ref{fig:abc1} and \ref{fig:abc2}
|
|
are identical. The actual presence of the multiple intersection zone existing in the concrete diagram, must be tested for
|
|
using area comparisions.
|
|
The test must use coordinates from the concrete diagram to perform an Area comparison.
|
|
\par
|
|
However, by only searching for them when a circular path is found, unnecessary searches are not undertaken.
|
|
This is the principal strength of this algorithm.
|
|
|
|
This will eliminate most non visible zones from requiring area searches.
|
|
|
|
\subsection{Zone Obscuration Test}
|
|
|
|
A zone enclosed by a member of its pure intersection, or enclosed completely will
|
|
be correctly cwdiscovered by the recursive and iterative process. However, it is possible that two or
|
|
more contours may combine to obscure a contour. This must be tested for, by
|
|
taking the area intersection of all contours comprising the potential zone,
|
|
and then subtracting all other areas. If any area remains, the zone is unobscured.
|
|
\par
|
|
Note that the obscuration test only need be applied within the pure intersection chain.
|
|
This is because a contour can only have a pure intersection chain or enclosure relationship with any other contour.
|
|
If the contour is enclosed it simply adds to the 'included ' contours.
|
|
Now only one (or more members in combination) of the same pure intersection chain can obscure a zone within it.
|
|
This reduces the number of obscuration tests to perform.
|
|
|
|
\par
|
|
If a zone is discovered and is not obscured, it is deemed a visible zone see equations \ref{available1},\ref{available2},\ref{available3},\ref{available4}.
|
|
|
|
|
|
\section{ Graph Traversal and Algorithm Development }
|
|
|
|
Using the rules from the previous section and some dry runs of the
|
|
procedure an efficient recursive algorithm has been developed.
|
|
Essentially, a spanning tree is created originating from each contour
|
|
in the diagram, using the pure intersection relationships.
|
|
|
|
\subsection { Traversal of a Simple Pure Intersection Chain }
|
|
|
|
Starting with the pure intersection chain in figure \ref{fig:pic}, with the contour A.
|
|
Note that create zone -or- CZ means that the potential zone, with intersecting elements
|
|
is checked for enclosure by all contours in the diagram with an enclosing relationship, and then
|
|
specifically created as an Area\cite{javaarea}, and this Area object is checked for
|
|
existence and for obscuration. NP means a repeated path, where zones rediscovered will not be
|
|
re-investigated as area objects.
|
|
\par
|
|
Iteration 1. Contour A
|
|
$$
|
|
\begin{array}{lclcl}
|
|
A & & & CZ & A \\
|
|
A & \stackrel{pi}\longrightarrow & B & CZ & A \cap B \\
|
|
B & \stackrel{pi}\longrightarrow & C & CZ & B \cap C\\
|
|
C & \stackrel{pi}\longrightarrow & D & CZ & C \cap D \\
|
|
\end{array}
|
|
$$
|
|
Note the individual case of A is considered, and then on through the chain of pure intersections.
|
|
Contour A is now marked as processed.
|
|
Next contour B is processed.
|
|
\par
|
|
Iteration 2. Contour B
|
|
$$
|
|
\begin{array}{lclcl}
|
|
B & & & CZ & B\\
|
|
B & \stackrel{pi}\longrightarrow & A & NP \\
|
|
B & \stackrel{pi}\longrightarrow & C & CZ & B \cap C \\
|
|
C & \stackrel{pi}\longrightarrow & D & CZ & C \cap D\\
|
|
\end{array}
|
|
$$
|
|
Iteration 3. Contour C
|
|
$$
|
|
\begin{array}{lclcl}
|
|
C & & & CZ & C\\
|
|
C & \stackrel{pi}\longrightarrow & B & NP & \\
|
|
C & \stackrel{pi}\longrightarrow & D & CZ & C \cap D \\
|
|
\end{array}
|
|
$$
|
|
|
|
Iteration 4. Contour D
|
|
$$
|
|
\begin{array}{lclcl}
|
|
D & & & CZ & D \\
|
|
D & \stackrel{pi}\longrightarrow & C & NP \\
|
|
\end{array}
|
|
$$
|
|
|
|
Using these iterations we have found all seven zones and only rediscovered
|
|
three during the iterations.
|
|
Only simple area searches were used to build the relation tables,
|
|
also for a brute for concrete area compare method,
|
|
we would have had to examine $ 2^{N} $ or 16 searches with an average of 4 compares (64 operations).
|
|
|
|
\subsection { Traversal of a Pure Intersection Chain with Implicit Enclosure }
|
|
|
|
The iteration for the diagram in figure \ref{fig:picwaie} is very similar to that
|
|
above, but the zones $ D \cup E \cup C $ would be created because of checking for enclosure within
|
|
a pure intersection chain, and $ C \cup E $ following the pure intersection path
|
|
$ C \; \stackrel{pi}\longrightarrow \; D $ and the standalone zone $ E $.
|
|
|
|
|
|
\subsection { Traversal of a Pure Intersection Chain with Circular Reference }
|
|
|
|
Here we encounter the possibility of a multiple intersection zone. The {\em CZ} action
|
|
must find the intersection, and then check that it really exists using area compares.
|
|
The iteration chain then for Figure \ref{fig:abc1} or \ref{fig:abc2}.
|
|
|
|
\par
|
|
Iteration 1. Contour A
|
|
$$
|
|
\begin{array}{lclcl}
|
|
A & & & CZ & A \\
|
|
A & \stackrel{pi}\longrightarrow & B & CZ & A \cap B \\
|
|
B & \stackrel{pi}\longrightarrow & C & CZ & B \cap C \\
|
|
C & \stackrel{pi}\longrightarrow & A & CZ & C \cap A \\
|
|
C & \stackrel{pi}\longrightarrow & C & CZ & B \cap C \cap A \\
|
|
\end{array}
|
|
$$
|
|
Note the last iteration has discovered a circular path. The CZ action will now investigate the presence
|
|
of the $ B \cap C \cap A $ intersection and register it if it is in the concrete diagram.
|
|
|
|
Iteration 2. Contour B.
|
|
$$
|
|
\begin{array}{lclcl}
|
|
B & & & CZ & B \\
|
|
B & \stackrel{pi}\longrightarrow & A & NP & \\
|
|
B & \stackrel{pi}\longrightarrow & C & NP & \\
|
|
\end{array}
|
|
$$
|
|
|
|
Iteration 3. Contour C.
|
|
$$
|
|
\begin{array}{lclcl}
|
|
C & & & CZ & C \\
|
|
C & \stackrel{pi}\longrightarrow & A & NP & \\
|
|
C & \stackrel{pi}\longrightarrow & B & NP & \\
|
|
\end{array}
|
|
$$
|
|
So, using the unordered relationships, determined by individual area compares, we are able to determine the
|
|
7 visible zones by graph traversal, and one Area check (for $ A \cup B\cup C $).
|
|
|
|
\section{Algorithm Efficiency : Iteration Comparisons against Brute Force Method}
|
|
\label{complexity}
|
|
The examples show in diagram form here use 8 contours per diagram. Using the Brute force method these would require $N 2^{N}$ or
|
|
2048 area compares to determine all visible zones.
|
|
By duplicating the structures, iteration values can be calculated for 16 contour diagrams of the same visual complexity.
|
|
|
|
\subsection { Simple pairs of contours}
|
|
|
|
The simple diagram, shown in figure \ref{fig:simple}, consists of four
|
|
overlapping pairs of contours. To determine the
|
|
enclosure and pure intersection relations, two cross products of area
|
|
searches are required. Thus $2 \;N^{2} $, i.e. 128 searches.
|
|
\par
|
|
Zones derived from single or pair contours do not need to be checked for existence. The pure intersection
|
|
relation has already determined the pair zone exists and a single contour always creates a zone.
|
|
Checking is required for obscuration is not required because none of the contours belong to a pure intersection.
|
|
\par
|
|
The total number of area compares/operations is therefore $2.64 \equiv 128 $
|
|
\par
|
|
As a general case, for extrapolating larger diagrams of the same pattern
|
|
|
|
\equation
|
|
\label{gensp}
|
|
2.N^{2}
|
|
\endequation
|
|
|
|
|
|
|
|
\begin{figure}
|
|
\vskip 6cm
|
|
\special{psfile=fzd/simple.ps hoffset=0 voffset=-10 hscale=40 vscale=40}\caption[Simple Pure Intersection Pairs]{
|
|
Simple Pure Intersection Pairs
|
|
\label{fig:simple}}
|
|
\end{figure}
|
|
|
|
\subsection {Two Venn 3 totally Enclosed Once : a more Complex Diagram}
|
|
|
|
The second diagram, see figure \ref{fig:tev3}, contains two Venn 3 configurations each enclosed by a contour.
|
|
Breaking this down, we have two single zones (i.e. G and H). These do not belong to a pure intersection chain and therefore require tests no for obscuration.
|
|
\par
|
|
Examining the two Venn3 structures, these require an existance check for the tripple intersection ( 3 area operations ). As the number of
|
|
contours to check for obscuration against it is 0, they do not obscuration testing.
|
|
\par
|
|
The 3 pairs require obscuration testing with the other countour. Thus 2 area operations to construct the shape of the zone,
|
|
and 1 area operation to test for obscuration, thus 3 per pair.
|
|
\par
|
|
The three single zones in the pure intersection require 1 area operation to construct the shape of the contour, and two to test for obscuration.
|
|
Thus 3 per pair.
|
|
|
|
|
|
|
|
This diagram therefore requires $128 + 2.(9 + 9) \equiv 146 $ area compares.
|
|
|
|
|
|
|
|
\equation
|
|
\label{genev3}
|
|
2.N^{2} + \frac{N}{4}.(18)
|
|
\endequation
|
|
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 585 410,keepaspectratio=true]{fzd/tripples.jpg}
|
|
% tripples.jpg: 813x569 pixel, 100dpi, 20.65x14.45 cm, bb=0 0 585 410
|
|
\caption{Two Enclosed Venn 3}
|
|
\label{fig:tev3}
|
|
\end{figure}
|
|
%
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/tripples.ps hoffset=0 voffset=-10 hscale=40 vscale=40}\caption[Two Enclosed Venn 3]{
|
|
% Two Enclosed Ven 3
|
|
% \label{fig:tev3}}
|
|
% \end{figure}
|
|
|
|
|
|
\subsection {Extrapolating for N Contour Diagrams}
|
|
|
|
Duplicating the structures in the diagrams in figures \ref{fig:tev3} and \ref{fig:simple},
|
|
and using the general
|
|
case equations (\ref{genev3} and \ref{gensp}), a plot of area searches required
|
|
against diagram complexity
|
|
can be drawn. These graphs clearly shows that the fzd method efficiency increases with the
|
|
number of contours in a diagram.
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 414 308,keepaspectratio=true]{fzd/perf1.jpg}
|
|
% perf1.jpg: 575x428 pixel, 100dpi, 14.60x10.87 cm, bb=0 0 414 308
|
|
\caption{Performace from 0 to 8 contours}
|
|
\label{fig:perf1}
|
|
\end{figure}
|
|
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/perf1.ps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Performance Comparison]{
|
|
% Perfomance from 0 to 8 contours
|
|
% \label{fig:perf1}}
|
|
% \end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=200pt,bb=0 0 414 308,keepaspectratio=true]{fzd/perf2.jpg}
|
|
% perf2.jpg: 575x428 pixel, 100dpi, 14.60x10.87 cm, bb=0 0 414 308
|
|
\caption{Performace from 8 to 64 contours}
|
|
\label{fig:perf2}
|
|
\end{figure}
|
|
|
|
% \begin{figure}
|
|
% \vskip 6cm
|
|
% \special{psfile=fzd/perf2.ps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Performance Comparison]{
|
|
% Performance from 8 to 64 contours
|
|
% \label{fig:perf2}}
|
|
% \end{figure}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% ALREADY DONE ! Obscuration is now only looked at within pure intersection chains.
|
|
% 05AUG2006
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
% \subsection { A general Case and Algorithm efficiency plotted against number of Contours }
|
|
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% \subsection {Further Optimisation}
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% If sections of the diagram can be identified that do not interact (enclose or purely intersect with other parts)
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% these can be analysed separately and reduce the $I$ and $O$ values. For instance in the Venn3 enclosed once,
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% the tests for existence and obscuration, need only be applied within each of the two identical
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% (in the abstract sense) separate structures. This would make the set G, discussed in equation \ref{seteq}
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% half the size of the number of contours in the diagram. Work on identifying separate areas and the
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% potential for nesting Euler diagrams is discussed in \cite{nied}.
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% \equation
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% \label{gopt}
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% \begin{array}{l}
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% G_{optimal} \; \stackrel{\Delta}{=} \; \forall \; C \; \bullet \; a \; X \; \forall \; C \; \bullet \; b \\
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% \; \bullet (a \stackrel{pi}\longrightarrow b) \; \vee \; (a \stackrel{enc}\longrightarrow b) \; \vee \;
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% (b \stackrel{enc}\longrightarrow a) \\
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%:wq
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%:wq \end{array}
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% \endequation
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% The optimal G sets are those within the diagram connected by any pure intersection or enclosure relationship.
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% This is expressed formally in equation \ref{gopt}.
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% \parsep=0pt
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% \itemsep=0pt
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% \parskip=3pt
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%
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% \small
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% \begin{thebibliography}{119}
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%
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%
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%
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% \bibitem{javaapp1} Gosling et. all {\em The Java Application Programming Interface Vol I : Addisson Wesley : ISBN 0 201 63453 8 }
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%
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% \bibitem{javaapp2} Gosling et. all {\em The Java Application Programming Interface Vol II : Addisson Wesley : ISBN 0 201 63459 7 }
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%
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% \bibitem{fft} Robert D Stum, Donald E Kirk {\em Discrete Systems and Digital Signal Processing : Addison-Wesley : ISBN 0-201-09518-1 }
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%
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% \bibitem{sem} Jim Woodcock, Martin Loomes {\em Software Engineering Mathematics : Pitman : ISBN 0-273-02673-9 }
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%
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% \bibitem{alggraph} Alan Gibbons {\em Algorithmic Graph Theory : Cambridge University Press : ISBN 0-521-28881-9 }
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%
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%
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% \bibitem{gtl} Cadwell {\em http://www.utm.edu/cgi-bin/caldwell/tutor/departments/math/graph/intro}
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%
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% \bibitem{dmnt} R Garnier, J Taylor {\em Discrete Mathematics for New Technology : IoP : ISBN 0-7503-0135-X }
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%
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% \bibitem{dmfss} D C Ince {\em An Introduction to Discrete Mathematics, Formal System Specification and Z : Oxford : ISBN 0-19-853836-7 }
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%
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% \bibitem{wiki} R.P. Clark et. all {\em http://en.wikipedia.org/wiki/Euler\_diagram : Wikipedia Euler Diagram reference }
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%
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% \bibitem{nied} J.A. Flower, J Howse, J Taylor {\em Nesting in Euler Diagrams } http://www.cmis.brighton.ac.uk/Research/vmg/papers/GTVMT02.pdf
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%
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% \bibitem{cried} Howse J, Stapleton G, Flower J, Taylor J {\em Corresponding regions in Euler diagrams} http://www.cmis.brighton.ac.uk/Research/vmg/papers/D2K2HSFT.pdf
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%
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% \bibitem{geneuler} Jean Flower, John Howse {\em Generating Euler Diagrams :Proceedings of Diagrams 2002, Callaway Gardens, Georgia, April 2002, Springer Verlag } http://www.it.bton.ac.uk/research/vmg/VisualModellingGroup.html
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%
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% \bibitem{javaarea} Sun Micro Systems {\em http://java.sun.com/j2se/1.3/docs/api/java/awt/geom/Area.html }
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%
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% \end{thebibliography}
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%
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%\end{document}
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%\theend
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