Higraphs: Visualising Information complex non-quantitative, - - PowerPoint PPT Presentation

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Higraphs: Visualising Information complex non-quantitative, - - PowerPoint PPT Presentation

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Higraphs: Visualising Information complex non-quantitative, structural topological, not geometrical Euler Venn diagrams (Jordan curve: inside/outside): enclosure, intersection graphs (nodes, edges: binary relation );

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SLIDE 1

Higraphs: Visualising Information

  • complex
  • non-quantitative, structural
  • topological, not geometrical
  • Euler

– Venn diagrams (Jordan curve: inside/outside): enclosure, intersection – graphs (nodes, edges: binary relation); hypergraphs

David Harel. On Visual Formalisms. Communications of the ACM. Volume 31, No. 5. 1988. pp. 514 - 530.

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 1/22

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SLIDE 2

Venn diagrams, Euler circles

R P Q B A

  • topological notions (syntax):

enclosure, exclusion, intersection

  • Used to represent (denote) mathematical set operations:

union, difference, intersection

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 2/22

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SLIDE 3

Hypergraphs

a b c d e f g h i a graph a hypergraph

  • topological notion (syntax): connectedness
  • Used to represent (denote) relations between sets.
  • Hyperedges: non longer binary relation (⊆ X ×X):

⊆ 2X (undirected), ⊆ 2X ×2X (directed).

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 3/22

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SLIDE 4

Higraphs: combining graphs and Venn diagrams

  • sets + cartesian product
  • hypergraphs

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 4/22

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SLIDE 5

Blobs: set inclusion, not membership

A D E

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 5/22

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SLIDE 6

Unique Blobs (atomic sets, no intersection)

B A C P Q R S U E K L M N O P W T V F D X

  • atomic blobs are identifiable sets
  • other blobs are union of enclosed sets (e.g.,

K = L∪M ∪N ∪O ∪P)

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 6/22

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SLIDE 7
  • empty space meaningless, identify intersection (e.g., N = K ∩W)

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 7/22

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SLIDE 8

Unordered Cartesian Product: Orthogonal Components

B A C P Q R S U E K L M N O P W X Y T V F D G H

K = G×H = H ×G = (L∪M)×(N ∪O ∪P)

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 8/22

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SLIDE 9

Meaningless syntactic constructs

A C D B

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 9/22

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SLIDE 10

Simple Higraph

J

blobs

  • rthogonal

components

A B D E C F L I H G K M

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 10/22

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SLIDE 11

Induced Acyclic Graph (blob/orth comp alternation)

  • h

c j k f i e d l b OR level (blob level) AND level (orthogonal component level) OR level AND level OR level g a m

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 11/22

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SLIDE 12

Adding (hyper) edges

A P Q R U K L M N P W X Y T V F D S O C B E

  • hyperedges
  • attach to contour of any blob
  • inter-level possible (e.g., denote global variables binding)

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 12/22

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SLIDE 13

Clique Example

A B C D E

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 13/22

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SLIDE 14

Clique: fully connected semantics

A B C D E

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 14/22

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SLIDE 15

Entity Relationship Diagram (is-a)

WORKS FOR EMPLOYEES IS A PAID ON IS A PILOTS CAN FLY AIRCRAFT SALARIES DATES SECRETARIES

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 15/22

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SLIDE 16

Higraph version of E-R diagram

employees

  • thers

pilots for paid

  • n

dates months can fly equipment aircraft nuts bolts arrived in years ... ... secretaries works salaries

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 16/22

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SLIDE 17

Extending the E-R diagram

employees for paid

  • n

can fly secretaries

  • thers

pilots men women works married

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 17/22

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SLIDE 18

Formally (syntax)

A higraph H is a quadruple H = (B,E,σ,π) B: finite set of all unique blobs E: set of hyperedges ⊆ X ×X, ⊆ 2X, ⊆ 2X ×2X The subblob (direct descendants) function σ σ : B → 2B σ0(x) = {x}, σi+1 =

  • y∈σi(x)

σ(y), σ+(x) =

+∞

  • i=1

σi(x)

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 18/22

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SLIDE 19

Subblobs+ cycle free x ∈ σ+(x) The partitioning function π associates equivalence relationship with x π : B → 2B×B Equivalence classes πi are orthogonal components of x π1(x),π2(x),...,πkx(x) kx = 1 means a single orthogonal component (no partitioning) Blobs in different orthogonal components of x are disjoint ∀y,z ∈ σ(x) : σ+(y)∩σ+(z) = ∅ unless in the same equivalence class

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 19/22

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SLIDE 20

Simple Higraph

J

blobs

  • rthogonal

components

A B D E C F L I H G K M

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 20/22

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SLIDE 21

Induced Orthogonal Components

B = {A,B,C,D,E,F,C,G,H,I,J,K,L,M} E = {(I,H),(B,J),(L,C)} ρ(A) = {B,C,H,J},ρ(G) = {H,I},ρ(B) = {D,E},ρ(C) = {E,F}, ρ(J) = {K,L,M} ρ(D) = ρ(E) = ρ(F) = ρ(H) = ρ(I) = ρ(K) = ρ(L) = ρ(M) = ∅ π(J) = {(K,K),(K,L),(L,L),(L,K),(M,M)} Induces equivalence classes π1(J) = {K,L} and π2(J) = {M}, . . . These are the orthogonal components

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 21/22

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SLIDE 22

Higraph applications (add specific meaning)

  • 1. E-R diagrams
  • 2. data-flow diagrams (activity diagrams)

edges represent (flow of) data

  • 3. inheritance
  • 4. Statecharts

Hans Vangheluwe hv@cs.mcgill.ca Higraphs 22/22