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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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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 2Andrew Fish, University of Brighton, UK

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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…? Andrew Fish, University of Brighton, UK3

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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. 4Andrew Fish, University of Brighton, UK

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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 Andrew Fish, University of Brighton, UK

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Euler Diagram Example LHS: 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}} A B C D {A,B} {B} {B,C} {C} {D} ∅ 6

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Euler Diagram Problems: static Concrete EDAbstract ED/ Set System 7Andrew Fish, University of Brighton, UK Abstraction Generation

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Euler Diagram Problems: dynamic Concrete EDAbstract ED 8Andrew Fish, University of Brighton, UK Abstraction Generation Transformations

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Euler Diagram Problems: dynamic Concrete EDAbstract ED 9Andrew Fish, University of Brighton, UK Abstraction Generation Transformations Wellformedness conditions

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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”. 10Andrew Fish, University of Brighton, UK

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Fast Diagram Interpretation Given a wellformed concrete Euler diagram d: 1.compute the abstract Euler diagram for d A B 4 Zones: {A} {A,B} {B} ∅ 11Andrew Fish, University of Brighton, UK

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Fast Diagram Interpretation Given 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 12Andrew Fish, University of Brighton, UK

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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 13Andrew Fish, University of Brighton, UK

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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 14Andrew Fish, University of Brighton, UK

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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. 15Andrew Fish, University of Brighton, UK

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Weakly reducible Euler Diagrams 16Andrew Fish, University of Brighton, UK Reducible Weakly Reducible Completely Reducible WF Euler diagrams ? ?

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Weakly reducible Euler Diagrams 17Andrew Fish, University of Brighton, UK Reducible Weakly Reducible Completely Reducible WF Euler diagrams ? ?

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Weakly reducible Euler Diagrams 18Andrew Fish, University of Brighton, UK Reducible Weakly Reducible Completely Reducible WF Euler diagrams ? ? Link to Proof

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Weakly reducible Euler Diagrams 19Andrew Fish, University of Brighton, UK Reducible Weakly 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 component 6 intersection points 1+6+1=8 zones 2 component 2 intersection points 2+2+1=5 zones 3 component 4 intersection points 3+4+1=8 zones 20

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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 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 21Andrew Fish, University of Brighton, UK

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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 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 22Andrew Fish, University of Brighton, UK

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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 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 23Andrew Fish, University of Brighton, UK

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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) 24Andrew Fish, University of Brighton, UK

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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 25Andrew Fish, University of Brighton, UK

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Marked Points A B C D E 26Andrew Fish, University of Brighton, UK 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.

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Marked Points A B C D E 27Andrew Fish, University of Brighton, UK 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.

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Marked Points A B C D E 28Andrew Fish, University of Brighton, UK 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.

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Marked Points A B C D E 29Andrew Fish, University of Brighton, UK 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.

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Marked Points A B C D E 30Andrew Fish, University of Brighton, UK 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.

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Marked Points A B C D E 31Andrew Fish, University of Brighton, UK 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.

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Ellipse Addition A B C D d Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram 32Andrew Fish, University of Brighton, UK

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Ellipse Addition A B C D d Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram Z d ={ , {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {C,D}, {A,B,C}} 33Andrew Fish, University of Brighton, UK

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Ellipse Addition d is a wellformed diagram A B C D d Z d ={ , {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {C,D}, {A,B,C}} Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram 34Andrew Fish, University of Brighton, UK

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Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram Ellipse Addition A B C D E x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 d Compute E's relationship with d Z d ={ ,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}} Cont(E)= Over(E)={A,B,C} Inter(E)={x 1,x 2,x 3,x 4,x 5,x 6 } 35Andrew Fish, University of Brighton, UK

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Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram Ellipse Addition A B C D E x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 d Is d +E wellformed ? By using Thm 1 Z d ={ ,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}} Cont(E)= Over(E)={A,B,C} Inter(E)={x 1,x 2,x 3,x 4,x 5,x 6 } 36Andrew Fish, University of Brighton, UK

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Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram Ellipse Addition Compute Split zones Z d ={ ,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}} Cont(E)= Over(E)={A,B,C} Inter(E)={x 1,x 2,x 3,x 4,x 5,x 6 } Each arc splits exactly one zone A B C D E d x2x2 x3x3 x4x4 x5x5 x6x6 x1x1 37Andrew Fish, University of Brighton, UK

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Compute Split zones Split Zones={ , {C}, {B,C},{B},{A,B},{A}} Z d ={ ,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}} Cont(E)= Over(E)={A,B,C} Inter(E)={x 1,x 2,x 3,x 4,x 5,x 6 } Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram Ellipse Addition A B C D E d x2x2 x3x3 x4x4 x5x5 x6x6 x1x1 Each arc splits exactly one zone 38Andrew Fish, University of Brighton, UK

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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 relationshi p 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 E E E E d + E is a wellformed diagram Ellipse Addition Z d ={ ,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}} Cont(E)= Over(E)={A,B,C} Inter(E)={x 1,x 2,x 3,x 4,x 5,x 6 } A B C D E d x2x2 x3x3 x4x4 x5x5 x6x6 x1x1 39Andrew Fish, University of Brighton, UK

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Update marked points Points x 2 and x 3 are swapped with y 1 and y 2, respectively. For instance, y 1 previously marking {C} in d now marks {C,E}, whilst {C} is marked by x 2. Draw a new contour E Compute E's relationshi p 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 E E E E d + E is a wellformed diagram Ellipse Addition A B C D E x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 d y1y1 y2y2 Z d ={ ,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}} Cont(E)= Over(E)={A,B,C} Inter(E)={x 1,x 2,x 3,x 4,x 5,x 6 } 40Andrew Fish, University of Brighton, UK

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Update covered zones Zones {A,C} and {A,B,C} are not split and their marking points, y 3 and y 4, 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 relationshi p 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 E E E E d + E is a wellformed diagram Ellipse Addition A B C D E x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 d y1y1 y2y2 y3y3 y4y4 Z d ={ ,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}} Cont(E)= Over(E)={A,B,C} Inter(E)={x 1,x 2,x 3,x 4,x 5,x 6 } 41Andrew Fish, University of Brighton, UK

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Ellipse Addition d+E is a wellformed diagram Z d+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 relationshi p 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 E E E E d + E is a wellformed diagram A B C D E 42Andrew Fish, University of Brighton, UK

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Ellipse Addition d+E is a wellformed diagram Z d+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 relationshi p 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 E E E E d + E is a wellformed diagram A B C D E d x2x2 x3x3 x4x4 x5x5 x6x6 x1x1 43Andrew Fish, University of Brighton, UK Skip Ellipse Deletion

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Ellipse Removal d=d+E Z d ={ , {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 Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C 44Andrew Fish, University of Brighton, UK

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Ellipse Removal d=d+E Z d ={ , {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 Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram C C C C C x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 45Andrew Fish, University of Brighton, UK

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Ellipse Removal d is a wellformed diagram Z d ={ , {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 Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C 46Andrew Fish, University of Brighton, UK

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Ellipse Removal Compute C ’s relationship with d-C Z d ={ , {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)={z 1,z 2,z 3,z 4,z 5,z 6 } Remove a contour C Compute C's relationship with d-C Is d - C wellformed ? Remove C from d Compute Split zones Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C z2z2 z3z3 z4z4 z5z5 z6z6 z1z1 47Andrew Fish, University of Brighton, UK

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Ellipse Removal Is d-C wellformed? Z d ={ , {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)={z 1,z 2,z 3,z 4,z 5,z 6 } Remove a contour C Compute C's relationship with d-C Is d - C wellformed ? Remove C from d Compute Split zones Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C z2z2 z3z3 z4z4 z5z5 z6z6 z1z1 Only disconnected zones need to be checked. 48Andrew Fish, University of Brighton, UK

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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 Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C z2z2 z3z3 z4z4 z5z5 z6z6 z1z1 Z d ={ , {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)={z 1,z 2,z 3,z 4,z 5,z 6 } Each arc splits exactly one zone 49Andrew Fish, University of Brighton, UK

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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 Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C z2z2 z3z3 z4z4 z5z5 z6z6 z1z1 Z d ={ , {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)={z 1,z 2,z 3,z 4,z 5,z 6 } Each arc splits exactly one zone 50Andrew Fish, University of Brighton, UK

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Ellipse Removal Remove marked points that belong to C Z d ={ , {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)={z 1,z 2,z 3,z 4,z 5,z 6 } Remove a contour C Compute C's relationship with d-C Is d - C wellformed ? Remove C from d Compute Split zones Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C z2z2 z3z3 z4z4 z5z5 z6z6 z1z1 51Andrew Fish, University of Brighton, UK

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Ellipse Removal Remove marked points that belong to C Z d ={ , {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)={z 1,z 2,z 3,z 4,z 5,z 6 } Remove a contour C Compute C's relationship with d-C Is d - C wellformed ? Remove C from d Compute Split zones Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C 52Andrew Fish, University of Brighton, UK

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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 Removal Z d ={ , {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)={z 1,z 2,z 3,z 4,z 5,z 6 } Remove a contour C Compute C's relationship with d-C Is d - C wellformed ? Remove C from d Compute Split zones Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram A B C D E C C C C C 53Andrew Fish, University of Brighton, UK

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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 Merge Zones Update Marked Points Update covered Zones Cont(E) Over(E) Inter(E) Reject A no yes d is a wellformed diagram C C d - C is a wellformed diagram C C C C C Z d-C ={ , {A}, {B}, {E}, {D}, {A,B}, {A,E}, {B,E}, {A,B,E},} A B D E 54Andrew Fish, University of Brighton, UK

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Complexity Ellipse addition and removal algorithms are similar and their complexity is 55Andrew Fish, University of Brighton, UK

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Complexity Draw a new contour E Compute E's relationshi p 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 no yes d is a wellformed diagram E E E E E d + E is a wellformed diagram O(|C|) O(|C| log |C|)O(|C|) O(|Z|) 56 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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Complexity Draw a new contour E Compute E's relationshi p 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 no yes d is a wellformed diagram E E E E E d + E is a wellformed diagram O(|C|) O(|C| log |C|)O(|C|) O(|Z|) 57 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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Complexity Draw a new contour E Compute E's relationshi p 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 no yes d is a wellformed diagram E E E E E d + E is a wellformed diagram O(|C|) O(|C| log |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). 58

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Complexity Draw a new contour E Compute E's relationshi p 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 no yes d is a wellformed diagram E E E E E d + E is a wellformed diagram O(|C|) O(|C| log |C|)O(|C|) O(|Z|) 59 Add E to split zones

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Complexity Draw a new contour E Compute E's relationshi p 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 no yes d is a wellformed diagram E E E E E d + E is a wellformed diagram O(|C|) O(|C| log |C|)O(|C|) O(|Z|) 60 Iterate through Inter(E) and perform set membership check

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Complexity Draw a new contour E Compute E's relationshi p 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 no yes d is a wellformed diagram E E E E E d + E is a wellformed diagram O(|C|) O(|C| log |C|)O(|C|) O(|Z|) 61 Iterate through non-split zone set and check if their marked point is in interior(E)

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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|)) f(|C|) = time needed to check if a given zone is present in d + E 62

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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… 63Andrew Fish, University of Brighton, UK

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“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 64Andrew Fish, University of Brighton, UK

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Bookmark Tags 65Andrew Fish, University of Brighton, UK

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EulerVC: skip slides; demo toolskip slidesdemo tool 66

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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)Symmetric Venn(5) with Ellipses Future plans: – Handle NWF cases by extending the use of marked ptsNWF – General resource handling (web pages, photos, files...) – Integration with EulerView idea (for larger scale)EulerView – Investigate diagram classes 82Andrew Fish, University of Brighton, UK

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