interactive visual classification with euler diagrams
DESCRIPTION
Interactive Visual Classification with Euler Diagrams. Gennaro Cordasco , Rosario De Chiara Università degli Studi di Salerno, Italy . Andrew Fish * University of Brighton, UK. *funded by UK EPSRC grant EP/E011160: Visualization with Euler Diagrams. Overview . Classification Problem - PowerPoint PPT PresentationTRANSCRIPT
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Interactive Visual Classification with Euler Diagrams
Gennaro Cordasco,Rosario De Chiara
Università degli Studi di Salerno, Italy
Andrew Fish*
University of Brighton, UK
*funded by UK EPSRC grant EP/E011160: Visualization with Euler Diagrams.
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Andrew Fish, University of Brighton, UK
Overview
• Classification Problem– related Euler Diagram applications
• Diagram Abstraction Problem• Concepts needed– e.g. weakly reducible, marked points, …
• On-line algorithms – complexity analysis
• EulerVC application demo
2
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Andrew Fish, University of Brighton, UK 3
Classification Problem
• Resource classification is often challenging for users, and commonly hierarchical classifications are not sufficient for their needs.
• Free-form tagging approaches provide a flat space, utilising different tagging and visualisation mechanisms.
• What about using non-hierarchical classification structures, and what about the use of (Euler) diagrams…?
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Andrew Fish, University of Brighton, UK
Related Euler Diagram Applications
• LHS [Inria]: Visualizing the numbers of documents matching a query from a library database, facilitating query modifications.
• RHS [Salerno]: File organisation with VennFS allows the user to draw Euler Diagrams in order to organize files within categories.
4
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Andrew Fish, University of Brighton, UK
The main problem
• To compute the set of zones associated to a given collection of curves
• Zones are not so easy to describe, they can be non-convex, non-simply connected,…
• How to update the zone set when we add curves?
5
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6
Euler Diagram ExampleLHS: A concrete diagram with 4 contours (simple closed
curves in plane) and 6 zones (regions inside a set of contours and outside the rest).
RHS: Depicts the abstract diagram/set system, which is the abstraction of the LHS: d= {A,B,C,D}, {∅,{B},{C},{D},{A,B},{B,C}}
ABC
D{A,B}
{B} {B,C}
{C} {D}
∅
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Andrew Fish, University of Brighton, UK 7
Euler Diagram Problems: static
Concrete EDAbstract ED/ Set System
Abstraction
Generation
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Andrew Fish, University of Brighton, UK 8
Euler Diagram Problems: dynamic
Concrete EDAbstract ED
Abstraction
Generation
TransformationsTransformations
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Andrew Fish, University of Brighton, UK 9
Euler Diagram Problems: dynamic
Concrete EDAbstract ED
Abstraction
Generation
TransformationsTransformations
Wellformedness conditions
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Andrew Fish, University of Brighton, UK 10
ED wellformedness conditions • contours are simple closed curves,
each with a single, unique label; • contours meet transversely (so no
tangential meetings or concurrency) and at most two contours meet at a single point;
• zones cannot be disconnected– i.e. each minimal region is a zone.
• Or, if you prefer… wellformed Euler diagrams can be thought of as “simple Euler diagrams”; i.e. simple Venn diagrams where some regions of intersection can be empty/missing”.
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Andrew Fish, University of Brighton, UK 11
Fast Diagram InterpretationGiven a wellformed concrete Euler diagram d:1. compute the abstract Euler diagram for d
A
B
4 Zones:{A}{A,B}{B}∅
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Andrew Fish, University of Brighton, UK 12
Fast Diagram InterpretationGiven a wellformed concrete Euler diagram d:1. compute the abstract Euler diagram for d 2. evaluate if the addition of a new contour or removal of an
existing contour yields a wellformed diagram (and update the abstract diagram accordingly).
A
B
A
B
C
C
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Andrew Fish, University of Brighton, UK 13
• On-line approach– We consider the diagram construction using the
natural operations of contour addition/removal.
Fast Diagram Interpretation
A
B
A
B
A
C
B
C
Timeline
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Andrew Fish, University of Brighton, UK 14
• On-line approach– We consider the diagram construction using the
natural operations of contour addition/removal.– What class of diagrams is this?
Fast Diagram Interpretation
A
B
A
B
A
C
B
C
Timeline
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Andrew Fish, University of Brighton, UK 15
Weakly reducible Euler Diagrams• Reducible Venn diagrams [e.g. see Ruskey 97]
– the removal of one of its curves yields a Venn diagram
• Reducible Euler diagrams– the removal of one of its curves yields a WF Euler diagram
• Completely reducible Euler diagrams – There is a sequence of curve removals that yields the empty
ED through WF diagrams (or a sequence of curve additions from the empty ED).
• Weakly reducible Euler diagrams – There is a sequence of curve removals and additions that
yields the empty ED through WF diagrams.
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Andrew Fish, University of Brighton, UK 16
Weakly reducible Euler Diagrams
ReducibleWeakly Reducible
Completely Reducible
WF Euler diagrams
?
?
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Andrew Fish, University of Brighton, UK 17
Weakly reducible Euler Diagrams
ReducibleWeakly Reducible
Completely Reducible
WF Euler diagrams
?
?
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Andrew Fish, University of Brighton, UK 18
Weakly reducible Euler Diagrams
ReducibleWeakly Reducible
Completely Reducible
WF Euler diagrams
?
?
Link to Proof
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Andrew Fish, University of Brighton, UK 19
Weakly reducible Euler Diagrams
ReducibleWeakly Reducible
Completely Reducible
WF Euler diagrams
?
?
Link to Proof
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Wellformedness online verification• Theorem 1: Let d = C, Z, be a weakly reducible Euler
diagram, then |Z| = ip(d) + cc(d) + 1 where – cc(d) denotes the number of components– ip(d) denotes number of intersection points
1 component6 intersection points 1+6+1=8 zones
2 component2 intersection points 2+2+1=5 zones
3 component4 intersection points 3+4+1=8 zones
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Andrew Fish, University of Brighton, UK 21
Wellformedness online verification• Theorem 1: Let d = C, Z, be a weakly reducible Euler
diagram, then |Z| = ip(d) + cc(d) + 1 where – cc(d) denotes the number of components– ip(d) denotes number of intersection points
• Proof: by induction for single component case– Use that the addition of a new contour generates x > 1 intersection points splitting the new contour into x segments, each of which splits a different zone (otherwise the component was not connected) and so we have x new zones.
A
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Andrew Fish, University of Brighton, UK 22
Wellformedness online verification• Theorem 1: Let d = C, Z, be a weakly reducible Euler
diagram, then |Z| = ip(d) + cc(d) + 1 where – cc(d) denotes the number of components– ip(d) denotes number of intersection points
• Corollary of Proof: The addition of a simple closed curve A to a weakly reducible diagram d which only meets d in transverse intersections at points which are not intersection points of d yields a WF Euler diagram iff d+A satisfies the “zones condition” above.
A
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Andrew Fish, University of Brighton, UK 23
Wellformedness online verification• Theorem 1: Let d = C, Z, be a weakly reducible Euler
diagram, then |Z| = ip(d) + cc(d) + 1 where – cc(d) denotes the number of components– ip(d) denotes number of intersection points
• Corollary of Proof: The addition of a simple closed curve A to a weakly reducible diagram d which only meets d in transverse intersections at points which are not intersection points of d yields a WF Euler diagram iff d+A satisfies the “zones condition” above.
• The point(s): – We have enough points to “mark” the zone set– Can check wellformedness after curve addition by counting
A
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Andrew Fish, University of Brighton, UK 24
Euler Diagram Abstraction
For our interface, we want both:– Quick construction method– Quick interpretation method
So, we consider the use of ellipses.Given two ellipses A and B, one can quickly find:
– their intersection points (in particular, since in a wellformed diagram tangential points are not allowed, we will have 0, 2 or 4 intersection points);
– their relationship (that is, if they overlap, if one is contained in the other or if they are disjoint)
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Andrew Fish, University of Brighton, UK 25
Marked Points• We can represent each zone of d using a single marked
point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the
closure of z.
A
B
C
D E
2( )m x m R ( ) ( )m x cl zm
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Andrew Fish, University of Brighton, UK 26
Marked Points
A
B
C
D E
• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the
closure of z.
2( )m x m R ( ) ( )m x cl zm
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Andrew Fish, University of Brighton, UK 27
Marked Points
A
B
C
D E
• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the
closure of z.
2( )m x m R ( ) ( )m x cl zm
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Andrew Fish, University of Brighton, UK 28
Marked Points
A
B
C
D E
• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the
closure of z.
2( )m x m R ( ) ( )m x cl zm
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Andrew Fish, University of Brighton, UK 29
Marked Points
A
B
C
D E
• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the
closure of z.
2( )m x m R ( ) ( )m x cl zm
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Andrew Fish, University of Brighton, UK 30
Marked Points
A
B
C
D E
• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the
closure of z.
2( )m x m R ( ) ( )m x cl zm
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Andrew Fish, University of Brighton, UK 31
Marked Points
A
B
C
D E
• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the closure of z. And we can update this set of marked points appropriately upon the addition or removal of curves.
2( )m x m R ( ) ( )m x cl zm
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Andrew Fish, University of Brighton, UK 32
Ellipse Addition
A
B
C
D
d
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
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Andrew Fish, University of Brighton, UK 33
Ellipse Addition
A
B
C
D
d
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Zd={, {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {C,D}, {A,B,C}}
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Andrew Fish, University of Brighton, UK 34
Ellipse Additiond is a wellformed diagram
A
B
C
D
d
Zd={, {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {C,D}, {A,B,C}}
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
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Andrew Fish, University of Brighton, UK 35
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Ellipse Addition
A
B
C
D E
x1
x2
x3
x4x5
x6
d
Compute E's relationship with d
Zd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}
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Andrew Fish, University of Brighton, UK 36
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Ellipse Addition
A
B
C
D E
x1
x2
x3
x4x5
x6
d
Is d +E wellformed ?
By using Thm 1
Zd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}
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Andrew Fish, University of Brighton, UK 37
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Ellipse AdditionCompute Split zones
Zd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}
Each arc splits exactly one zone
A
B
C
D E
dx2
x3
x4x5
x6
x1
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Andrew Fish, University of Brighton, UK 38
Compute Split zones
Split Zones={, {C}, {B,C},{B},{A,B},{A}}
Zd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Ellipse Addition
A
B
C
D E
dx2
x3
x4x5
x6
x1
Each arc splits exactly one zone
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Andrew Fish, University of Brighton, UK 39
Generate new zones
Split Zones={, {C}, {B,C},{B},{A,B},{A}}New Zones={{E}, {C,E}, {B,C,E}, {B,E}, {A,B,E}, {A,E}}
Draw a new
contour E
Compute E's
relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Ellipse AdditionZd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}
A
B
C
D E
dx2
x3
x4x5
x6
x1
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Andrew Fish, University of Brighton, UK 40
Update marked points
Points x2 and x3 are swapped with y1 and y2, respectively. For instance, y1 previously marking {C} in d now marks {C,E}, whilst {C} is marked by x2.
Draw a new
contour E
Compute E's
relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Ellipse Addition
A
B
C
D E
x1
x2
x3
x4x5
x6
d
y1
y2
Zd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}
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Andrew Fish, University of Brighton, UK 41
Update covered zones
Zones {A,C} and {A,B,C} are not split and their marking points, y3 and y4, are in interior(E), so these zones need to be updated and become {A,C,E} and {A,B,C,E}.
Draw a new
contour E
Compute E's
relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
Ellipse Addition
A
B
C
D E
x1
x2
x3
x4x5
x6
d
y1
y2y3
y4
Zd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}
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Andrew Fish, University of Brighton, UK 42
Ellipse Additiond+E is a wellformed diagram
Zd+E={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}
Draw a new
contour E
Compute E's
relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
A
B
C
D E
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Andrew Fish, University of Brighton, UK 43
Ellipse Additiond+E is a wellformed diagram
Zd+E={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}
Draw a new
contour E
Compute E's
relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagram
A
B
C
D E
dx2
x3
x4x5
x6
x1
Skip Ellipse Deletion
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Andrew Fish, University of Brighton, UK 44
Ellipse Removald=d+E
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
C
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Andrew Fish, University of Brighton, UK 45
Ellipse Removald=d+E
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}
A
B
C
D E
d
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
C
C
C
C
Cx1
x2
x3
x4x5
x6
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Andrew Fish, University of Brighton, UK 46
Ellipse Removald is a wellformed diagram
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
C
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Andrew Fish, University of Brighton, UK 47
Ellipse RemovalCompute C’s relationship with d-C
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
Cz2
z3
z4
z5
z6
z1
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Andrew Fish, University of Brighton, UK 48
Ellipse RemovalIs d-C wellformed?
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
Cz2
z3
z4
z5
z6
z1
Only disconnected zones need to be checked.
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Andrew Fish, University of Brighton, UK 49
Compute Split zones
Split Zones={ ,{B}, {E}, {A,E}, {B,E}, {A,B,E}}
Ellipse Removal
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
Cz2
z3
z4
z5
z6
z1
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}
Each arc splits exactly one zone
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Andrew Fish, University of Brighton, UK 50
Merge zones
Ellipse Removal
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
Cz2
z3
z4
z5
z6
z1
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}
Each arc splits exactly one zone
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Andrew Fish, University of Brighton, UK 51
Ellipse RemovalRemove marked points that belong to C
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
Cz2
z3
z4
z5
z6
z1
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Andrew Fish, University of Brighton, UK 52
Ellipse RemovalRemove marked points that belong to C
Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}
Remove a
contour C
Compute C's
relationship with d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
C
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Andrew Fish, University of Brighton, UK 53
Update covered zones
Zone {C,D} is not split by C and its marked point is in interior(C), so it needs to be updated and it becomes {D}.
Ellipse RemovalZd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}
Remove a
contour C Compute
C's relationship with
d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
A
B
C
D E
C
C
C
C
C
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Andrew Fish, University of Brighton, UK 54
d-C is a wellformed diagram
Ellipse Removal
Remove a
contour C Compute
C's relationship with
d-C
Is d - C wellformed
?
Remove C from d Compute
Split zones
MergeZones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
C
C
d -C is a wellformed diagram
C
C
C
C
C
Zd-C={, {A}, {B}, {E}, {D}, {A,B}, {A,E}, {B,E}, {A,B,E},}
A
B
D E
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Andrew Fish, University of Brighton, UK 55
Complexity
• Ellipse addition and removal algorithms are similar and their complexity is
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56
Complexity
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagramO(|C|)
O(|C|)
O(|C| log |C|)O(|C|)
O(|C|)
O(|Z|)
For each other curve, compute intersection points with E; if curve C disjoint then check if marked point for C is in interior (E)
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57
Complexity
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagramO(|C|)
O(|C|)
O(|C| log |C|)O(|C|)
O(|C|)
O(|Z|)
Check no tangential/triple points created. Check WF using the number of intersection points of E with other curves (Thm 1)
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58
Complexity
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagramO(|C|)
O(|C|)
O(|C| log |C|)O(|C|)
O(|C|)
O(|Z|)
a) Each arc splits exactly 1 zone
b) # arcs = O(|C|)c) Consecutive
arcs split zones which differ by one exactly contour, so we order the points of inter(E).
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59
Complexity
Draw a new
contour E Compute
E's relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagramO(|C|)
O(|C|)
O(|C| log |C|)O(|C|)
O(|C|)
O(|Z|)
Add E to split zones
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60
Complexity
Draw a new
contour E
Compute E's
relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagramO(|C|)
O(|C|)
O(|C| log |C|)O(|C|)
O(|C|)
O(|Z|)
Iterate through Inter(E) and perform set membership check
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61
Complexity
Draw a new
contour E
Compute E's
relationship with
d
Is d +E wellformed
?
Add E to d
Compute Split zones
Generate new
Zones
Update Marked Points
Update covered Zones
Cont(E)Over(E
)Inter(E)
Reject A
noyes
d is a wellformed diagram
E
EE
E
E
d +E is a wellformed diagramO(|C|)
O(|C|)
O(|C| log |C|)O(|C|)
O(|C|)
O(|Z|)
Iterate through non-split zone set and check if their marked point is in interior(E)
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62
A naive approach
Draw a new
contour E
A new contour
is added
Duplicate each zone
Check Zones
d is an Euler diagram
E
E
O(1)
O(|Z|)
O(|Z|f(|C|))
O(|Z|f(|C|))
f(|C|) = time needed to check if
a given zone is present in d + E
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Andrew Fish, University of Brighton, UK 63
EulerVC
• As a proof of concept we have implemented the algorithms as general purpose Java library– And used it to build a small application which
allows users to use Euler Diagrams to classify internet bookmarks on their desktop, or on Delicious…
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Andrew Fish, University of Brighton, UK 64
“Delicious is a social bookmarking web service for storing, sharing, and discovering web bookmarks” (from Wikipedia)
– Users can tag each of their bookmarks with freely chosen index terms
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Andrew Fish, University of Brighton, UK 65
BookmarkTags
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66
EulerVC: skip slides; demo tool
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Andrew Fish, University of Brighton, UK 67
EulerVC
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Andrew Fish, University of Brighton, UK 68
EulerVC
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Andrew Fish, University of Brighton, UK 69
EulerVC
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Andrew Fish, University of Brighton, UK 70
EulerVC
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Andrew Fish, University of Brighton, UK 71
EulerVC
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Andrew Fish, University of Brighton, UK 72
EulerVC
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Andrew Fish, University of Brighton, UK 73
EulerVC
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Andrew Fish, University of Brighton, UK 74
EulerVC
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Andrew Fish, University of Brighton, UK 75
EulerVC
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Andrew Fish, University of Brighton, UK 76
EulerVC
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Andrew Fish, University of Brighton, UK 77
EulerVC
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Andrew Fish, University of Brighton, UK 78
EulerVC
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Andrew Fish, University of Brighton, UK 79
EulerVC
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Andrew Fish, University of Brighton, UK 80
EulerVC
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81
EulerVC
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Andrew Fish, University of Brighton, UK 82
Conclusion• Algorithms developed to solve the on-line diagram
abstraction problem.• EulerVC application constructed useful for:
– User classification of resources (requires user testing…)– Exploratory research of diagram properties like weak
reducibility (see Symmetric Venn(5) with Ellipses)
• Future plans: – Handle NWF cases by extending the use of marked pts– General resource handling (web pages, photos, files...) – Integration with EulerView idea (for larger scale)– Investigate diagram classes
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Andrew Fish, University of Brighton, UK 83
Blank