1.4 Euler Diagram Layout Techniques

Size: px

Start display at page:

Download "1.4 Euler Diagram Layout Techniques"

Maude Kelley
6 years ago
Views:

1 1.4 Euler Diagram Layout Techniques

2 Euler Diagram Layout Techniques: Overview Dual graph based methods Inductive methods Drawing with circles Including software demos. How is the drawing problem stated?

3 Zones A C z1 z3 z 2 z 4 z 5 z 6 B z 1 = z 2 = A z 3 = AC z 3 = ABC z 5 = AB z 6 = B An abstract description is a set of zone descriptions (simply called abstract zones, or just zones).

4 Euler Diagram Layout Techniques: Generation Problem Given an abstract description (a collection of zones), find an Euler diagram with that abstract description.

5 Question Euler Diagram Layout Techniques: Generation Problem Given an abstract description (a collection of zones), find an Euler diagram with that abstract description. Can you draw this? A, B, BC

6 Euler Diagram Layout Techniques: Generation Problem Question Given an abstract description (a collection of zones), find an Euler diagram with that abstract description. Can you draw this? Solution A, B, BC A B C

7 Dual Graph Generation D Convert to a vertexlabelled graph An embedding of D G Embed in a particular manner Construct any edgelabelled dual Transform G d Convert edges to curves SG* Remove edges labelled {} G*

8 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC {} Step 1 A B BC

9 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC {} A B A B C C A B BC Step 2

10 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC A C B Step 3

11 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC {} Step 1 A B BC

12 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC {} A {} B {} C A B BC Step 1b

13 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC {} {} B A B C A C A B BC {} {} {} Step 2

14 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC B C A {} {} Step 2b

15 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC B C A Step 2b

16 Dual Graph Generation Problem Draw a diagram with this abstraction using a dual graph method A, B, BC B C A Step 3

17 Dual Graph Generation Summary Create dual graph (red) Create dual of this graph Remove edges labelled {} to create Euler graph Convert Euler graph to Euler diagram

18 Dual Graph Generation Advantages Disadvantages Can draw any abstract description Capable of enforcing different collections of properties Computationally expensive Hard to identify duals that give rise to required properties Not all diagrams are aesthetically pleasing

19 Dual Graph Generation DEMO

20 Inductive Generation decompose D D=D n D n-1... D 1 D 0 generated D n ensure correct abstractions... generate D 0 add curves

21 Inductive Generation Problem Draw a diagram with this abstraction using an inductive method P, Q, PQ P P Q embed(d 0 ) embed(d 1 ) embed(d 2 )

22 Inductive Generation Adding curves using cycles P Q P Q P R Q R d 4 d 5 d 6

23 Inductive Generation Problem Adding curves using cycles: modifying the Euler dual P R Q P R Q d 7 S d 8

24 Inductive Generation Problem Adding curves using cycles: the hybrid graph P Q P Q P Q R d 9 d 10 d 11

25 Inductive Generation Creating the hybrid graph P R Q d 1 G 2 G 3 G 4 HG(d 1 )

26 Inductive Generation Method Summary Add one curve at a time Create hybrid graph Find appropriate cycle to add desired curve The properties of the cycle correspond to properties possessed by the resulting diagram

27 Inductive Generation P R Q P R Q Example d 12 d 13 Both diagrams have same abstraction, but only one possesses the simplicity property.

28 Inductive Generation Advantages Disadvantages Can draw any abstract description Easily capable of enforcing different collections of properties: easy to see from chosen cycle which properties are enforced Computationally expensive: can result in Hamiltonian cycles being sought Not all diagrams are aesthetically pleasing

29 Inductive Generation DEMO

30 Advantages Disadvantages Piercing Generation: Using Circles This approach draws diagrams with circles All diagrams drawn possess all of the six properties Diagrams are aesthetically pleasing Very efficient: polynomial time complexity Cannot draw some abstract descriptions

31 Piercing Generation: Using Circles abstract syntax A, B, AB, C, AC, BC, ABC, D, BD

32 Piercing Generation: Using Circles Question Is this drawable with circles? A, B, AB, C, AC, BC, ABC, D, BD, CD, BCD

33 Piercing Generation: Using Circles Question Answer Is this drawable with circles? A, B, AB, C, AC, BC, ABC, D, BD, CD, BCD Yes: Question How can we identify whether an abstract description is drawable with circles?

34 Piercing Generation: Using Circles Question Is this drawable with circles? A, B, AB, C, AC, BC, ABC, D, AD, BD, CD, ABD, ACD, BCD, ABCD

35 Piercing Generation: Using Circles Question Answer Is this drawable with circles? A, B, AB, C, AC, BC, ABC, D, AD, BD, CD, ABD, ACD, BCD, ABCD No, but it can be drawn with ellipses: A B C D

36 Single Piercings C contains two zones Removing C yields: Zones combine when C is removed. C is a single piercing of A. At the abstract level: A, B, AB, BC, ABC becomes A, B, AB, NOTE: BC and ABC differ only by A, the pierced curve lose the two abstract zones containing C.

37 Double Piercings D contains four zones. Removing D yields: Zones combine when D is removed. D is a double piercing of A and B. At the abstract level: A, B, AB, C, AC, BC, ABC, CD, ACD, BCD, ABCD becomes A, B, AB, C, AC, BC, ABC lose the four abstract zones containing D.

38 Building Diagrams from Piercings

39 Building Diagrams from Piercings: Disconnected Diagrams

40 Building Diagrams from Piercings: Choices of embedding

41 Building Diagrams from Piercings: Choices of embedding

42 Building Diagrams from Piercings: No choice of embedding We want to draw A, B, AB, C, AC, BC, ABC, CD, ACD, BCD, ABCD, E, AE, BE, ABE [C, D and E are double piercings of A and B] Cannot add E as a circle

43 Abstract Level: Inductive Definition An abstract description, d, is inductively pierced if 1 d has no curve labels, or 2 d has a base piercing, λ 1 and d with the zones containing λ 1 removed is inductively pierced, 3 d has a single piercing, λ 1 and d with the zones containing λ 1 removed is inductively pierced, or 4 d has a double piercing, λ 1, of λ 2 and λ 3 such that d with the zones containing λ 1 removed is inductively pierced, and a no other curve label in d is outside-associated with λ 2 and λ 3 or b exactly one other curve label, λ 4, in d is outside-associated with λ 2 and λ 3 and i the curve labels containing λ 1 are the same as those containing λ 4 and λ 2, or ii the curve labels containing λ 1 are the same as those containing λ 2 and λ 3.

44 Abstract Level: Inductive Definition Theorem Lemma If d is an inductively pierced abstract description then d is drawable with circles, satisfying all wfc, in polynomial time. If d is an inductively pierced abstract description with piercing curve label λ then d λ is inductively pierced.

45 Piercing Generation: Using Circles DEMO

Drawing Euler Diagrams with Circles: The Theory of Piercings

JOURNL OF L T E X CLSS FILES, VOL. 6, NO. 1, JNURY 2007 1 Drawing Euler Diagrams with Circles: The Theory of Piercings Gem Stapleton, Leishi Zhang, John Howse and Peter Rodgers bstract Euler diagrams are