Generating and Drawing Area-Proportional Euler and Venn

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Generating and Drawing Area-Proportional Euler and Venn Powered By Docstoc
					         Generating and Drawing
Area-Proportional Euler and Venn Diagrams

                 Stirling Chow

           Department of Computer Science
                University of Victoria


               Ph.D. Examination
                 May 10, 2007
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapters 1/2: Introduction

                                (B)irds            (C)arnivores
                 (A)nimals




                       152      13        8        64         11




                                                                       752




                                     (B)irds            (C)arnivores
                    (A)nimals




                             152       13      8    64       11




                                                                       752
Chapters 1/2: Basic Definitions

       c1
             {1}

            {1,2}                c3
c2                                           Euler† vs. Euler-like‡ Diagrams
                           {3}
                                             Connected vs. Disconnected
            {2}
                                             Concurrent vs. Non-concurrent
                                             Non-pairwise vs. Pairwise
  c1                                         Non-simple vs. Simple
                  c2                  c3     (Simple ≡ Non-concurrent & Pairwise)
              {2}                     {3}   † RWD: U. Brighton and U. Kent, UK
                                            ‡ AV Library: INA and INRIA, France
                   {1,2}
              {1}
Chapters 1/2: Problem Definitions

                             Set System and Items
                             S = {∅, {1}, {1, 2}, {2}, {3}}
                             X = {1, 2, 3}

     c1
           {1}               Euler Diagram Generation Problem (EDGP)
                              INPUT:       A set system S.
          {1,2}         c3
                              OUTPUT:      An Euler diagram with
c2
                  {3}                      regions S.
          {2}
                             Area-Proportional EDGP (ω-EDGP)
                              INPUT:       A set system S and
                                           function ω : S\{∅} → R+ .
                              OUTPUT:      An ω-proportional
                                           Euler diagram with regions S.
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapter 3: ω-proportional n-Euler Diagrams, n ≤ 3

low blood
 pressure
                             nausea
                                       Two circles with bisection†

            0.45   0.10   0.20




                                      † Solution to ω-EDGP.
Chapter 3: ω-proportional n-Euler Diagrams, n ≤ 3

      c1                      Two circles with bisection†
                              Three circles with hillclimbing
                              (w/ Peter Rodgers)
                  d13
           d12
                             † Solution to ω-EDGP.
                 d23
                        c3



 c2
Chapter 3: ω-proportional n-Euler Diagrams, n ≤ 3

                                              Two circles with bisection†
            c1   {1}                          Three circles with hillclimbing
                                        c3
                                              (w/ Peter Rodgers)
                       {1,3}      {3}
                                              Three orthogonal rectangles/6-gons‡
 c2                    {1,2,3} {2,3}         † Solution to ω-EDGP.
            {1,2}                            ‡ Solution to ω-EDGP with fullset.
      {2}
Chapter 3: Simple 3-Venn Diagrams with Convex Curves


  Theorem
  There exists a weight function ω for which there is no
  ω-proportional simple 3-Venn diagram whose curves are convex.
                         c1




                 c2


                                 c3
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapters 4/5: Euler Duals



                   c1 , c2


                  {1,2}
                                        c3
                             {1,2,3}

                                        {3,4} {4}

                                  {3}
                                                    c4
                                         {3,5}

                                             {5}
                                                    c5
Chapters 4/5: Euler Duals




                {1,2}



                        {1,2,3}         {3,4}    {4}

                                                       ∅
                                  {3}
                                            {3,5}


                                                {5}
Chapters 4/5: Euler Duals




                {1,2}



                        {1,2,3}         {3,4}    {4}

                                                       ∅
                                  {3}
                                            {3,5}


                                                {5}
Chapters 4/5: Euler Duals




                {1,2}



                        {1,2,3}         {3,4}    {4}

                                                       ∅
                                  {3}
                                            {3,5}


                                                {5}
Chapters 4/5: Euler Duals




                {1,2}



                        {1,2,3}         {3,4}    {4}

                                                       ∅
                                  {3}
                                            {3,5}


                                                {5}
Chapters 4/5: Connectivity Graphs


  Connectivity Graph
  Let S be a set system on the items X . A graph G is said to be a
  connectivity graph for S if and only if the following properties hold
  for G :
    1   G is a connected planar graph on S,
    2   G is loop-free, and
    3   for each item x ∈ X ,
          1   the subgraph Gx induced by the vertices containing x is
              connected, and
          2   the subgraph Gx induced by the vertices not containing x is
              connected.

  An Euler dual is a connectivity graph for S.
Chapters 4/5: Existence Conditions



  Theorem
  Let S be a set system. There exists an Euler diagram representing
  S if and only if there is a connectivity graph for S that has no
  parallel edges.

  Corollary
  The EDDP is decidable and there is an exponential time algorithm
  for constructing connectivity graphs.

  Note
  The proofs of the existence conditions are constructive.
Chapters 4/5: Existence Conditions




  Euler Diagram Generation Problem (EDGP)
   INPUT:     A set system S.
   OUTPUT:    A simple connectivity graph for S.

  Euler Diagram Decision Problem (EDDP)
   INPUT:     A set system S.
   OUTPUT:    “Yes” if there is a simple connectivity graph for S;
              otherwise, “No”.
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapter 6: Expressiveness of Diagram Types
        Set Systems

            Euler−like Diagrams
                               all Jordan curves
                                                            C

                                                   C


                                  Connected
                                 Euler Diagrams
                       B         Non−concurrent     B
                           Simple
                       B                                B

                                  A                  all curves
                                                   non−concurrent
                                             diagram
                                            connected
                      all intersections
                         pairwise
Chapter 6: Expressiveness of Diagram Types

      Set Systems

                         Euler−like,              Euler−like,
                     all intersections             diagram
                          pairwise                connected




                                         Simple

               all Jordan curves


                                                    all curves
                                                  non−concurrent
Chapter 6: Expressiveness of Diagram Types

                                                                                                                                      {1,2,6,7,8}


                                {1}                 {1,2}               {1}      {1,3}       {2,3}           {2}
                                                                                                                          {5,8,10}    {1,2,3,4,5}        {3,6,9}

                     {1}

                  {1,2,3}             Set Systems
                                                                                                                                     {1,4,7,9,10}
                                                        Euler−like,                 Euler−like,
                                                    all intersections                diagram
                                                         pairwise                   connected

            {1,2}
           {1,2,3}
           {1,2,3,4}
                                                                                                                                             {1}
            {3,4}                                                       Simple
                                             all Jordan curves
                                                                                                                                     {1,2}           {1,3}
                                                                                   all curves
                                                                                 non−concurrent

                                                                                                                               {2}           {2,3}           {3}




            {1}                                                               {1}
                                                                                                  {1,2}

                            {1,2}
                                                                                                  {2,3}
                                                                                                  {1,2,3}


                            {2,3}
                            {1,2,3}
                                                                                     {1,4}                  {1,5}



                                                                              {4}             {4,5}                 {5}
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapter 7: Computational Complexity of Existence




  Euler Diagram Decision Problem (EDDP)
   INPUT:     A set system S.
   OUTPUT:    “Yes” if there is a simple connectivity graph for S;
              otherwise, “No”.

  Theorem
  The EDDP is NP-complete.
Chapter 7: Previous Hypergraph Planarity Result



  Vertex-Based Hypergraph Planarity Problem (HPLANAR)
   INPUT:     A hypergraph H = (V , E ).
   OUTPUT:    “Yes” if there is a planar graph G on V
              for which each hyperedge e ∈ E (where e ⊆ V ),
              induces a connected subgraph of G ;
              otherwise, “No”.

  Theorem
  The HPLANAR is NP-complete (Johnson and Pollak ’87).
Chapter 7: Hypergraphs to Euler-like Diagrams



  Partial Connectivity Graph
  Let S be a set system on the items X . A graph G is said to be a
  partial connectivity graph for S if and only if the following
  properties hold for G :
    1   G is a connected planar graph on S,
    2   G is loop-free, and
    3   for each item x ∈ X ,
          1   the subgraph Gx induced by the vertices containing x is
              connected, and
          2   the subgraph Gx induced by the vertices not containing x is
              connected.
Chapter 7: Previous Euler-like Diagram Result




  Euler-like Diagram Decision Problem (weak-EDDP)
   INPUT:      A set system S.
   OUTPUT:     “Yes” if there is a partial connectivity graph for S;
               otherwise, “No”.

  Theorem
  The weak-EDDP is NP-complete (Verroust and Viaud ’04).
Chapter 7: Our Result


  Theorem
  The EDDP is NP-complete.

  NPC Reduction
     Extension of Johnson and Pollak’s gadget transformation from
     Hamilton Path on cubic 3-connected planar graphs
      Structured in terms of set systems rather than hypergraphs
      Additional structure to ensure Gx connectivity

  Basic Idea
  Construct set system S based on cubic 3-connected planar graph
  G so that any connectivity graph for S contains a subgraph
  homeomorphic to G .
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapter 8: Prime Factorization of Diagrams



                                                    ∅



                                                                         c8
                                          c1
                                                                        c4
                        c1                                    *
                        c2
                        c3
                             c8
                                                                              c9   c10
  c6   c5                                               c5
                   c4        c9   c10     c2
                                                                  c12

                                                                                         c11
  c7
            c1 2
                                    c11        c6        c7
                                          c3
Chapter 8: Prime Factorization Algorithm




  Factorization Problem
   INPUT:        A set system S on the items X .
   OUTPUT:       A decomposition of S into factorization tree.

  Theorem
  There is an O(|X |2 |S| + |X |3 ) algorithm for solving the
  Factorization Problem.
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapter 9: Redrawing Monotone Euler Diagrams

  Theorem
  Any monotone n-Euler diagram can be redrawn area-proportionally
  in O(n|V |) time.




       c1
                  c4




  c2                    c3
Chapter Overview
      Area-Proportionality                         Topology
          (ω-EDGP)           Chapters 1/2          (EDGP)
                             Background

          Chapters 3                             Chapters 4/5
       Preliminary Cases                     Graph-Theoretic Model
            (n ≤ 3)
                                                   Chapters 6
          Chapters 9                         Diagram Expressiveness
     Diagrams with Fullset
           (any n)
                                                  Chapters 7
                                            Computational Complexity


                                                  Chapter 8
                                              Composite Diagrams

                             Chapters 10
                             Conclusion
Chapter 10: Conclusion

  Main Contributions
      First systematic approach to area-proportional drawing;
      unique convex curve non-existence proof
      Concise Euler diagram existence conditions
      First mathematically-based analysis of expressiveness;
      understanding functional and aesthetic properties
      First NPC result related to Euler diagrams (not Euler-like
      diagrams); motivating composite Euler diagram heuristic
      All drawing algorithms implemented and in use by third parties