visualisation of overlapping sets and clusters with euler diagrams

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  • Universit Bordeaux 1

    cole doctorale de mathmatiques et informatique

    Doctorat en informatique

    Visualisation of Overlapping Sets and Clusters

    with Euler Diagrams

    Candidate: Thesis Directors:Paolo Simonetto David Auber

    Guy Melanon

    December 2011

  • In the loving memory of John,Giovanna and Sante.

  • Abstract

    In this thesis, we propose a method for the visualisation of overlapping sets and offuzzy graph clusterings based on Euler diagrams.

    Euler diagrams are probably the most intuitive and most used method to depictsets in which elements can be shared. Such a powerful visualisation metaphor couldbe an invaluable visualisation tool, but the automatic generation of Euler diagramsstill presents many challenging problems. First, not all instances can be drawn usingstandard Euler diagrams. Second, most existing algorithms focus on diagrams ofmodest dimensions while real-world applications typically features much larger data.Third, the generation process must be reliable and reasonably fast.

    In this thesis, we describe an extended version of Euler diagrams that can be pro-duced for every input instance. We then propose an automatic procedure for thegeneration of such diagrams that specifically target large input instances. Finally,we present a software implementation of this method and we describe some outputexamples generated on real-world data.

    Rsum

    Dans cette thse, nous proposons une mthode pour la visualisation densembles che-vauchant et de bas sur les diagrammes dEuler.

    Les diagrammes dEuler sont probablement les plus intuitifs pour reprsenter demanire schmatique les ensembles qui partagent des lments. Cette mtaphore vi-suelle est ainsi un outil puissant en termes de visualisation dinformation. Cependant,la gnration automatique de ces diagrammes prsente encore de nombreux problmesdifficiles. Premirement, tous les clustering chevauchants ne peuvent pas tre dessinesavec les diagrammes dEuler classiques. Deuximement, la plupart des algorithmesexistants permettent uniquement de reprsenter les diagrammes de dimensions mo-destes. Troisimement, les besoins des applications relles requirent un processusplus fiable et plus rapide.

    Dans cette thse, nous dcrivons une version tendue des diagrammes dEuler.Cette extension permet de modliser lensemble des instances de la classe des clusteringchevauchants. Nous proposons ensuite un algorithme automatique de gnration decette extension des diagrammes dEuler. Enfin, nous prsentons une implmentationlogicielle et des exprimentations de ce nouvel algorithme.

    v

  • Acknowledgements

    I am very grateful to the members of the examination panel of my thesis for theircomments and questions. In particular, I would like to thank Stephen Kobourov andJarke van Wijk, the reviewers, for their invaluable corrections and suggestions, andDavid Auber and Guy Melanon, my supervisors, for supporting and helping me inmany ways, even outside the research environment.

    I would like to express my gratitude to two good friends and major co-authors ofmy articles: Romain Bourqui and Daniel Archambault. I thank Daniel particularlyfor the invaluable tutoring he provided me on scientific writing and on the Englishlanguage, that deeply influenced the clarity of my publications.

    I would also like to thank the friends who reviewed part of my thesis. Amongthese, a special thanks to Andrew Collins and Frank Hopfgartner.

    Finally, the greatest thanks goes to my girlfriend, Karen Jespersen, for the dedica-tion she put in reviewing and correcting my writing. For this reason, she would reallyhave deserved to be mentioned as an author in every single publication, including thisthesis.

    Thanks a lot!

    vii

  • Contents

    1 Introduction 1

    2 Visualisation and Euler Diagrams 32.1 Visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2.1.1 The Human Eye and Perception . . . . . . . . . . . . . . . . . 52.1.2 Evolution of Visualisation . . . . . . . . . . . . . . . . . . . . . 82.1.3 Visualisation and Computer Science . . . . . . . . . . . . . . . 92.1.4 Reasons and Goals of Visualisation . . . . . . . . . . . . . . . . 112.1.5 Disciplines of Visualisation . . . . . . . . . . . . . . . . . . . . 13

    2.2 Euler Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 The Original Diagrams . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Modern Euler and Venn Diagrams . . . . . . . . . . . . . . . . 19

    2.3 Data Visualisation with Euler Diagrams . . . . . . . . . . . . . . . . . 212.3.1 Euler Diagrams and Perception . . . . . . . . . . . . . . . . . . 212.3.2 An Example of Data Exploration with Euler Diagrams . . . . . 222.3.3 Limitations of Euler and Venn Diagrams . . . . . . . . . . . . . 24

    3 Graph and Euler Diagram Theory 273.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    3.1.1 Sets, Multiset and Tuples . . . . . . . . . . . . . . . . . . . . . 293.1.2 Graphs and Their Classification . . . . . . . . . . . . . . . . . . 303.1.3 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.4 Relations Between Nodes and Edges . . . . . . . . . . . . . . . 313.1.5 Walks, Paths, Cycles and Distance . . . . . . . . . . . . . . . . 323.1.6 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.7 Trees and Forests . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.8 Complete, Bipartite Graphs and Subdivisions . . . . . . . . . . 353.1.9 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.10 Dual Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.11 Clustered Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    3.2 Graph Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.1 Foundations of Algorithmics . . . . . . . . . . . . . . . . . . . . 413.2.2 Aesthetics of a Graph Drawing . . . . . . . . . . . . . . . . . . 423.2.3 Graph Drawing Algorithms . . . . . . . . . . . . . . . . . . . . 44

    3.3 Euler Diagram Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.1 Clusters and Zones . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.2 Euler Diagrams and Regions . . . . . . . . . . . . . . . . . . . 513.3.3 Properties of Euler Diagrams . . . . . . . . . . . . . . . . . . . 53

    ix

  • x Contents

    3.3.4 Validity of an Euler Diagram . . . . . . . . . . . . . . . . . . . 553.3.5 Drawability of an Euler diagram . . . . . . . . . . . . . . . . . 57

    4 Algorithms for the Generation of Euler Diagrams 594.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    4.1.1 Well-Formed Euler Diagrams . . . . . . . . . . . . . . . . . . . 594.1.2 Standard Euler Diagrams . . . . . . . . . . . . . . . . . . . . . 634.1.3 Extended Euler Diagrams . . . . . . . . . . . . . . . . . . . . . 644.1.4 Relaxed Euler Diagrams . . . . . . . . . . . . . . . . . . . . . . 654.1.5 Specialities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.1.6 Methods with Analogies to Euler Diagrams . . . . . . . . . . . 71

    4.2 Euler Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2.1 The Generation Process . . . . . . . . . . . . . . . . . . . . . . 734.2.2 Comparison with Methods in the Literature . . . . . . . . . . . 74

    5 Automatic Generation of Euler Representations 775.1 Generation and Embedding of the Zone Graph . . . . . . . . . . . . . 78

    5.1.1 Indentification of the Expressed Zones . . . . . . . . . . . . . . 785.1.2 Insertion of the Zone Graph Edges . . . . . . . . . . . . . . . . 785.1.3 Embedding of the Zone Graph . . . . . . . . . . . . . . . . . . 84

    5.2 Generation and Improvement of the Grid Graph . . . . . . . . . . . . 865.2.1 Grid Graph Generation . . . . . . . . . . . . . . . . . . . . . . 885.2.2 Grid Graph Improvement . . . . . . . . . . . . . . . . . . . . . 92

    5.3 Depiction of the Cluster Regions . . . . . . . . . . . . . . . . . . . . . 945.3.1 Smooth Cluster Curves . . . . . . . . . . . . . . . . . . . . . . 945.3.2 Assignment of the Cluster Colours . . . . . . . . . . . . . . . . 995.3.3 Application of Textures . . . . . . . . . . . . . . . . . . . . . . 99

    6 Improvement of a Graph Layout 1016.1 PrEd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    6.1.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1.3 Force Computation . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1.4 Maximal Movement Computation . . . . . . . . . . . . . . . . 1056.1.5 Displacement of the Nodes . . . . . . . . . . . . . . . . . . . . 1086.1.6 Advantages and Disadvantages . . . . . . . . . . . . . . . . . . 109

    6.2 ImPrEd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2.1 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.2.3 Force and Movement Cooling . . . . . . . . . . . . . . . . . . . 1146.2.4 Surrounding Edges Computation . . . . . . . . . . . . . . . . . 1176.2.5 QuadTrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2.6 New Maximal Movement Rules . . . . . . . . . . . . . . . . . . 1256.2.7 Crossable and Flexible Edges . . . . . . . . . . . . . . . . . . . 1296.2.8 Weight of Nodes and Edges . . . . . . . . . . . . . . . . . . . . 132

    6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.3.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3.2 Execution Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3.3 Drawing Quality and Parameter Reliabi

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